## Trigonometry

Another mnemonic probably more common in Great Britain than the United States is "Tommy On A Ship Of His Caught A Herring."

From the Pythagorean theorem ,

It is therefore also true that

The trigonometric functions can be defined algebraically in terms of complex exponentials (i.e., using the Euler formula ) as

Hybrid trigonometric product/sum formulas are

Osborn's rule gives a prescription for converting trigonometric identities to analogous identities for hyperbolic functions .

For imaginary arguments,

For complex arguments,

The complex modulus also satisfies the curious identity

The only functions satisfying identities of this form,

## Explore with Wolfram|Alpha

More things to try:

- trigonometry
- triangle 2, 3, 5
- angle 50 degrees

## Referenced on Wolfram|Alpha

Cite this as:.

Weisstein, Eric W. "Trigonometry." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Trigonometry.html

## Subject classifications

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## What Is Trigonometry? Definition, Formulas & Applications

## In this article

Trigonometry is a branch of mathematics that deals with the study of triangles. It is sometimes informally referred to as "trig." In trigonometry, mathematicians study the relationships between the sides and angles of triangles. Right triangles, which are triangles with one angle of 90 degrees, are a key focus of study in this area of mathematics.

Trigonometry is widely taught in high schools, typically to students in their junior year. Trigonometry builds on basic math concepts learned in lower grade levels, and it can be a confusing subject to navigate. Trigonometry tutors can either help students get a better grasp on trigonometry concepts or challenge them to think outside of the box, depending on their academic level and understanding of trigonometry. Read on to find out more about the invention of trigonometry, the best way to master it, how tutors can help, and how trig is different from other math subjects.

## What Does Trigonometry Mean?

The word trigonometry originates from the Greek words trigonon and metron, which mean "triangle" and "measure," respectively. The basics of trigonometry may be even older, and may have been used in ancient Egypt. Essentially, trigonometry deals with triangles and how the sides and angle measurements of a triangle relate to one another.

There is a wide range of careers that require the use of trigonometry, including architect, astronaut, crime scene investigator, physicist, surveyor, engineer, and many more. Even if you aren’t planning to pursue a career in any of these fields, it's still important to understand basic trigonometry concepts. Trigonometry is a required class in most school districts, and if students don't do well in this subject, it can reflect poorly on their overall academic record. There are several reasons why trigonometry is currently taught in schools, and many ways in which trigonometry can be used and applied in later life.

## Where Does Trigonometry Come From?

Trigonometry has a long history, dating back to the ancient world. Initially, trigonometry was concerned with utilizing basic functions to use the known angle of a triangle in order to calculate the remaining angles. For example, if two side lengths of a triangle were given and the measure of the enclosed angle was known, the third side and the two other angles could be calculated. Trigonometry, both then and now, is primarily focused on the relationships between the angles of triangles.

The ancient Greeks formalized the first trigonometric functions, starting with Hipparchus of Bithynia, circa 150 B.C. Hipparchus considered every type of triangle, including spherical triangles, right-angled triangles, and planar triangles. While Hipparchus's interest in trigonometric functions was mainly related to astronomy, trigonometry in modern times can be applied to many real-world situations.

## What Is Trigonometry Used For?

As previously mentioned, trigonometry is often utilized in fields related to physics, chemistry, and engineering. Far from simply being a theoretical subject, there are numerous practical applications of trigonometry. Engineers in many industrial fields use trig in the course of their work. Other professionals that may use trigonometry include surveyors, architects, and pilots. Just as Hipparchus did over two thousand years ago, trigonometry is used in astronomy to measure the distance of nearby stars and to calculate stellar positions. Trigonometry also plays a key role in satellite navigation systems.

One real-life problem that can be solved using the rules of trigonometry is to work out the measurements of things or spaces that would be difficult to measure directly. For example, trigonometric functions can be used to calculate the heights of mountains, the quantity of water in a lake, and the square footage of an unusually-shaped piece of land. Trigonometry can even be used to help astronomers measure time accurately.

Trigonometry concepts are used in gaming and music, as well. Trigonometry plays a role in video game development by ensuring that games function properly. It’s used to write programs so that objects in the game can move. Trigonometry can also be utilized in the process of designing characters, sets, and objects for video games. In music, trigonometry measures the level or pitch of a sound wave or musical note.

Thus, although trigonometry is certainly a math concept, it can be applied to many different fields and tasks outside of mathematics. These are all real-life scenarios in which a person would use trigonometry, and they’re examples of why it’s important to learn how to apply trigonometry correctly.

## When Do You Learn Trigonometry?

Because trigonometry can be applied to careers in many different fields, it’s important to ensure that students have a good grasp of trig concepts. Trigonometry builds on basic math knowledge taught throughout a student’s education, including algebra and geometry concepts. Because trigonometry combines concepts from different math subjects, there is a specific order in which these subjects are taught.

Typically, students take Algebra I, Geometry, and then Algebra II or Trigonometry. Depending on the state, some schools offer students a choice of which math subject they’d like to study. A student’s ability to choose a specific math subject may also depend on their academic performance. Since trigonometry applies concepts taught in earlier grade levels and builds upon previous knowledge, it can be a tough subject to tackle.

## Can You Learn Trigonometry Online?

With the rise of online learning, there have been amazing advances in how virtual classrooms function. There are many ways that online education can help all learners to master trigonometry. By using educational apps and software, teachers can share their computer screens and show exactly how to solve math problems, just as they would use a whiteboard or chalkboard to work out problems for students in the classroom.

Online classes are often smaller than in-person classes, allowing teachers to focus on assisting students who need help with specific trigonometry concepts. Online education has developed significantly in the last few years and now offers a safe space for all types of learners and learning styles.

## Is Trigonometry Hard?

While math comes naturally to some, trigonometry can be a confusing topic for many people. When you consider that a trigonometry class may involve trigonometric functions, trigonometric values, trigonometric formulas, and much more, it can be overwhelming and confusing to keep all of these components straight. The best way to ensure that students are using tangent, sine, and cosine functions correctly is to have guidance from someone with a detailed understanding of trigonometry. This could be a classmate, parent, teacher, or tutor.

Even if a student can perform algebraic functions and has a good grasp of concepts in other branches of mathematics, they might struggle with certain elements of trigonometry, such as the trigonometric ratios sine (sin), cosine (cos), and tangent (tan). Regardless of whether a student learns in a face-to-face classroom or a virtual classroom, it can be very helpful to have one-on-one help from a tutor. That way, students can receive immediate feedback. A trigonometry tutor can help students feel confident about the new concepts they’re encountering and can reinforce what they’re learning by providing extra practice.

## How Learner Trigonometry Tutors Can Help

Trigonometry can be a daunting subject, especially for people who don’t naturally excel in math or science. For parents, it can be tough to expect their children to excel in every subject. It’s worth remembering that there are many different types of learners and many different learning styles. Some people learn best with visuals, some people learn best by hearing the information, and some people learn best when they can read and listen to the instructions for a specific task. Some students may benefit from the help of an online math tutor. Tutors can also help students who are performing well academically by offering extra practice before college prep tests or school exams.

Many education-related businesses claim to help students, but it can be difficult to distinguish between companies that see young people as dollar signs and those that genuinely want students to succeed. It can also be challenging to find a tutor who fits a family’s busy schedule. Luckily, there are platforms that offer excellent online tutors who are custom-matched to students through a rigorous screening process. One of the best known is Learner. Learner focuses on providing specific academic instruction that is tailored to each student, rather than supplying generic lessons to all students.

Learner is so confident that they can meet their customers’ tutoring needs that they have a 100% risk-free satisfaction guarantee. Learner’s innovative platform offers collaborative tools to boost engagement, feedback metrics that let students know where they stand, and session recordings so that students can review later. and Learner works across all devices. Learner is one of many online tutoring websites, but it’s one of the only platforms that uses personalized matching with world-class tutors to drive accelerated learning.

Need help with trigonometry? Speak with our academic advisor to get matched with a top online math tutor today!

## Frequently Asked Questions About Trigonometry

Is trigonometry geometry.

Geometry is the branch of mathematics that studies the properties and relations of points, lines, surfaces, and solids, whereas trigonometry focuses exclusively on angles and triangles. Geometry is often taught in schools before trigonometry.

## Is Trigonometry Precalculus?

Trigonometry is a prerequisite to precalculus courses because precalculus uses elements of trigonometry, algebra, and analytical geometry.

## Is Algebra 2 Trigonometry?

Algebra 2 does not teach the same math concepts as trigonometry. Algebra 2 covers topics such as linear equations, functions, quadratic equations, polynomials, radical expressions, inequalities, graphs, and matrices. Trigonometry, on the other hand, is concerned with the specific functions of angles and how they are applied to calculations.

## What Are Trigonometric Identities?

Trigonometric identities are equalities involving trigonometric functions. They are important elements in the study of triangles. Trigonometric identities include Pythagorean identities, reduction formulas, and cofunction identities. Often, a scientific calculator is used to solve trig problems such as trigonometric identities.

## What Are Trigonometric Functions?

Trigonometric functions describe the relationships between the angles and sides of a triangle. In modern mathematics, there are six main trigonometric functions, also called trigonometric formulas: sine, tangent, secant, cosine, cotangent, and cosecant. These functions describe the ratios of the sides of right triangles.

## What Is Sin in Trigonometry?

Sine (sin), cosine (cos), and tangent (tan) are the three basic functions used in trigonometry. They’re based on right-angled triangles. The sine of one angle of a right triangle is the ratio of the length of the side of the triangle opposite the angle to the length of the triangle’s hypotenuse.

## What Is Cos in Trigonometry?

Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions based on right-angled triangles. The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

## What Is the Tangent Function in Trigonometry?

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side of that angle.

## What Is the Pythagorean Theorem?

The Pythagorean theorem states that for a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

## Does Trigonometry Work with Non-Right Angles?

Typically, high school trigonometry courses focus on applying trigonometric functions such as sine, cosine, and tangent to right triangles (which have one right angle). More advanced college trigonometry could include these functions as well as the application of trigonometry to non-right angles.

## What Do Trigonometry Courses Cover?

Trigonometry courses typically cover topics such as the Pythagorean identity and how to use trigonometric functions such as sine and cosine to solve right triangles. More advanced courses may include the study of complex numbers, polar coordinates, De Moivre's Theorem, and Euler's Formula.

## What Comes After Trigonometry?

In most schools, the math curriculum follows a specific order. Typically, students take Algebra I, Geometry, Algebra II or Trigonometry, Precalculus, and Calculus. Some students may take honors courses, which means they will begin Algebra I in eighth grade, instead of ninth grade. This would allow them to take Calculus during their senior year, instead of Precalculus. The curriculum in private schools may differ from this order.

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## Related Articles

## How to Do Trigonometry: Simplifying the Problem

Trigonometry is all about triangles. From hypotenuses to tangents, here’s a handy guide on how to avoid confusion in trigonometry.

A trigonometry tutor is an individual who tutors another person or people in mathematics, specifically in the area of trigonometry. Trigonometry is a branch of mathematics that focuses on the study of triangles; most students learn this subject in high school.

## Trigonometry

Trigonometry is the branch of mathematics that deals with the relationship between ratios of the sides of a right-angled triangle with its angles. The ratios used to study this relationship are called trigonometric ratios, namely, sine, cosine, tangent, cotangent, secant, cosecant. The word trigonometry is a 16th century Latin derivative and the concept was given by the Greek mathematician Hipparchus.

Here in the below content, we shall understand the basics of trigonometry, the various identities-formulas of trigonometry, and the real-life examples or applications of trigonometry.

## Introduction to Trigonometry

Trigonometry is one of the most important branches in mathematics. The word trigonometry is formed by clubbing words 'Trigonon' and 'Metron' which means triangle and measure respectively. It is the study of the relation between the sides and angles of a right-angled triangle. It thus helps in finding the measure of unknown dimensions of a right-angled triangle using formulas and identities based on this relationship.

## Trigonometry Basics

Trigonometry basics deal with the measurement of angles and problems related to angles. There are three basic functions in trigonometry: sine, cosine, and tangent. These three basic ratios or functions can be used to derive other important trigonometric functions: cotangent, secant, and cosecant. All the important concepts covered under trigonometry are based on these functions. Hence, further, we need to learn these functions and their respective formulas at first to understand trigonometry.

In a right-angled triangle, we have the following three sides.

Perpendicular - It is the side opposite to the angle θ. Base - This is the adjacent side to the angle θ. Hypotenuse - This is the side opposite to the right angle .

## Trigonometric Ratios

There are basic six ratios in trigonometry that help in establishing a relationship between the ratio of sides of a right triangle with the angle. If θ is the angle in a right-angled triangle, formed between the base and hypotenuse , then

- sin θ = Perpendicular/Hypotenuse
- cos θ = Base/Hypotenuse
- tan θ = Perpendicular/Base

The value of the other three functions: cot, sec, and cosec depend on tan, cos, and sin respectively as given below.

- cot θ = 1/tan θ = Base/Perpendicualr
- sec θ = 1/cos θ = Hypotenuse/Base
- cosec θ = 1/sin θ = Hypotenuse/Perpendicular

## Trigonometric Table

The trigonometric table is made up of trigonometric ratios that are interrelated to each other – sine, cosine, tangent, cosecant, secant, cotangent. These ratios, in short, are written as sin, cos, tan, cosec, sec, cot, and are taken for standard angle values. You can refer to the trigonometric table chart to know more about these ratios.

## Important Trigonometric Angles

Trigonometric angles are the angles in a right-angled triangle using which different trigonometric functions can be represented. Some standard angles used in trigonometry are 0º, 30º, 45º, 60º, 90º. The trigonometric values for these angles can be observed directly in a trigonometric table. Some other important angles in trigonometry are 180º, 270º, and 360º. Trigonometry angle can be expressed in terms of trigonometric ratios as,

- θ = sin -1 (Perpendicular/Hypotenuse)
- θ = cos -1 (Base/Hypotenuse)
- θ = tan -1 (Perpendicular/Base)

## List of Trigonometric Formulas

There are different formulas in trigonometry depicting the relationships between trigonometric ratios and the angles for different quadrants. The basic trigonometry formulas list is given below:

1. Trigonometry Ratio Formulas

- sin θ = Opposite Side/Hypotenuse
- cos θ = Adjacent Side/Hypotenuse
- tan θ = Opposite Side/Adjacent Side
- cot θ = 1/tan θ = Adjacent Side/Opposite Side
- sec θ = 1/cos θ = Hypotenuse/Adjacent Side
- cosec θ = 1/sin θ = Hypotenuse/Opposite Side

2. Trigonometry Formulas Involving Pythagorean Identities

- sin²θ + cos²θ = 1
- tan 2 θ + 1 = sec 2 θ
- cot 2 θ + 1 = cosec 2 θ

3. Sine and Cosine Law in Trigonometry

- a/sinA = b/sinB = c/sinC
- c 2 = a 2 + b 2 – 2ab cos C
- a 2 = b 2 + c 2 – 2bc cos A
- b 2 = a 2 + c 2 – 2ac cos B

Here a, b, c are the lengths of the sides of the triangle and A, B, and C are the angle of the triangle.

The complete list of trigonometric formulas involving trigonometry ratios and trigonometry identities is listed for easy access. Here's a list of all the trigonometric formulas for you to learn and revise.

## Trigonometric Functions Graphs

Different properties of a trigonometric function like domain, range, etc can be studied using the trigonometric function graphs. The graphs of basic trigonometric functions- Sine and Cosine are given below:

The domain and range of sin and cosine functions can thus be given as,

- sin θ: Domain (-∞, + ∞); Range [-1, +1]
- cos θ: Domain (-∞ +∞); Range [-1, +1]

Click here to learn about the graphs of all trigonometric functions and their domain and range in detail- Trigonometric Functions

## Unit Circle and Trigonometric Values

Unit circle can be used to calculate the values of basic trigonometric functions- sine, cosine, and tangent . The following diagram shows how trigonometric ratios sine and cosine can be represented in a unit circle .

## Trigonometry Identities

In Trigonometric Identities, an equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved. In trigonometric identities , you will get to learn more about the Sum and Difference Identities.

For example, sin θ/cos θ = [Opposite/Hypotenuse] ÷ [Adjacent/Hypotenuse] = Opposite/Adjacent = tan θ

Therefore, tan θ = sin θ/cos θ is a trigonometric identity. The three important trigonometric identities are:

- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ

## Applications of Trigonometry

Throughout history, trigonometry has been applied in areas such as architecture, celestial mechanics, surveying, etc. Its applications include in:

- Various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, and many more.
- It is also helpful to find the distance of long rivers, measure the height of the mountain, etc.
- Spherical trigonometry has been used for locating solar, lunar, and stellar positions.

## Real-Life Examples of Trigonometry

Trigonometry has many real-life examples used broadly. Let’s get a better idea of trigonometry with an example. A boy is standing near a tree. He looks up at the tree and wonders “How tall is the tree?” The height of the tree can be found without actually measuring it. What we have here is a right-angled triangle, i.e., a triangle with one of the angles equal to 90 degrees. Trigonometric formulas can be applied to calculate the height of the tree, if the distance between the tree and boy, and the angle formed when the tree is viewed from the ground is given.

It is determined using the tangent function, such as tan of angle is equal to the ratio of the height of the tree and the distance. Let us say the angle is θ, then

tan θ = Height/Distance between object & tree Distance = Height/tan θ

Let us assume that distance is 30m and the angle formed is 45 degrees, then

Height = 30/tan 45° Since, tan 45° = 1 So, Height = 30 m

The height of the tree can be found out by using basic trigonometry formulas.

☛ Related Topics:

- What Is a Radian
- Trigonometric Ratios in Radians
- Tangent Function
- Heights and Distances

Important Notes on Trigonometry

- Trigonometric values are based on the three major trigonometric ratios: Sine, Cosine, and Tangent. Sine or sin θ = Side opposite to θ / Hypotenuse Cosine or cos θ = Adjacent side to θ / Hypotenuse Tangent or tan θ = Side opposite to θ / Adjacent side to θ

0°, 30°, 45°, 60°, and 90° are called the standard angles in trigonometry.

The trigonometry ratios of cosθ, secθ are even functions, since cos(-θ) = cosθ, sec(-θ) = secθ.

## Solved Examples on Trigonometry

Example 1: The building is at a distance of 150 feet from point A. Can you calculate the height of this building if tan θ = 4/3 using trigonometry?

The base and height of the building form a right-angle triangle. Now apply the trigonometric ratio of tanθ to calculate the height of the building.

In Δ ABC, AC = 150 ft, tanθ = (Opposite/Adjacent) = BC/AC 4/3 = (Height/150 ft) Height = (4×150/3) ft = 200ft

Answer: The height of the building is 200ft.

Example 2: A man observed a pole of height 60 ft. According to his measurement, the pole cast a 20 ft long shadow. Find the angle of elevation of the sun from the tip of the shadow using trigonometry.

Let x be the angle of elevation of the sun, then

tan x = 60/20 = 3 x = tan -1 (3) or x = 71.56 degrees

Answer: The angle of elevation of the sun is 71.56º.

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## Practice Questions on Trigonometry

Faqs on trigonometry, what is trigonometry in math.

Trigonometry is the branch of mathematics that deals with the study of the relationship between the sides of a triangle (right-angled triangle) and its angles. The relationship is presented as the ratio of the sides, which are trigonometric ratios. The six trigonometric ratios are sine, cosine, tangent, cotangent, secant, and cosecant.

## What are the Basics of Trigonometry?

Trigonometry basics deal with the measurement of angles and problems related to angles. There are six basic trigonometric ratios: sine, cosine, tangent, cosecant, secant and cotangent. All the important concepts covered under trigonometry are based on these trigonometric ratios or functions.

## What are the Applications of Trigonometry?

Trigonometry finds applications in different fields in our day-to-day lives. In astronomy, trigonometry helps in determining the distances of the Earth from the planets and stars. It is used in constructing maps in geography and navigation. It can also be used in finding an island's position in relation to the longitudes and latitudes. Even today, some of the technologically advanced methods which are used in engineering and physical sciences are based on the concepts of trigonometry.

## How do You do SOH CAH TOA in Trigonometry?

We use the "SOH CAH TOA" trick to memorize the relationship between trigonometric ratios easily. To remember them, remember the word " SOHCAHTOA "!

- SOH : S ine = O pposite / H ypotenuse
- CAH : C osine = A djacent / H ypotenuse
- TOA : T angent = O pposite / A djacent

## What Does θ Mean in Trigonometry?

θ in trigonometry is used as a variable to represent a measured angle. It is the angle between the horizontal plane and the line of sight from an observer's eye to an object above. θ can be referred to as the angle of elevation or angle of depression, depending upon the object's position, i.e, when the object is above the horizontal line, θ is called the angle of elevation , and for object's position below the horizontal line, it is called the angle of depression .

## What are Trigonometry Identities?

Trigonometry identities are equations of trigonometry functions that are always true. Trigonometry identities are often used not only to solve trigonometry problems but also to understand important mathematical principles and solve numerous math problems.

## What are the Six Basic Trigonometry Functions?

There are 6 trigonometry functions which are:

- Sine function
- Cosine function
- Tangent function
- Secant function
- Cotangent function
- Cosecant function

## What is the Reciprocal of sin in Trigonometry?

The sine function for angle θ in a right-angled triangle in trigonometry is given as, sinθ = perpendicular /hypotenuse. The reciprocal of sin function is given as cosecant function. Therefore, cosecθ = hypotenuse/perpendicular.

## How is Trigonometry Used in Real Life?

Trigonometry in real life is used in the naval and aviation industries. It also finds application in cartography (creation of maps). It can be used to design the inclination of the roof and the height of the roof in buildings etc.

## Who Invented Trigonometry?

Hipparchus(c. 190–120 BCE), also known as the "father of trigonometry", was the first to construct a table of values for a trigonometric function

- Math Article

## Trigonometry

Trigonometry is one of the important branches in the history of mathematics that deals with the study of the relationship between the sides and angles of a right-angled triangle. This concept is given by the Greek mathematician Hipparchus. In this article, we are going to learn the basics of trigonometry such as trigonometry functions, ratios, trigonometry table, formulas and many solved examples.

## What is Trigonometry?

Trigonometry is one of the most important branches in mathematics that finds huge application in diverse fields. The branch called “Trigonometry” basically deals with the study of the relationship between the sides and angles of the right-angle triangle. Hence, it helps to find the missing or unknown angles or sides of a right triangle using the trigonometric formulas, functions or trigonometric identities. In trigonometry, the angles can be either measured in degrees or radians. Some of the most commonly used trigonometric angles for calculations are 0°, 30°, 45°, 60° and 90°.

Trigonometry is further classified into two sub-branches. The two different types of trigonometry are:

- Plane Trigonometry
- Spherical Trigonometry

In this article, let us discuss the six important trigonometric functions, ratios, trigonometry table, formulas and identities which helps to find the missing angles or sides of a right triangle.

## Trigonometry Ratios-Sine, Cosine, Tangent

The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle.

Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse are referred to as the adjacent and opposite sides.

## Six Important Trigonometric Functions

The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas and considering the above figure. It is necessary to get knowledge about the sides of the right triangle because it defines the set of important trigonometric functions.

## Even and Odd Trigonometric Functions

The trigonometric function can be described as being even or odd.

Odd trigonometric functions: A trigonometric function is said to be an odd function if f(-x) = -f(x) and symmetric with respect to the origin.

Even trigonometric functions: A trigonometric function is said to be an even function, if f(-x) = f(x) and symmetric to the y-axis.

We know that

- Sin (-x) = – Sin x
- Cos (-x) = Cos x
- Tan (-x) = -Tan x
- Csc (-x) = – Csc x
- Sec (-x) = Sec x
- Cot (-x) = -Cot x

Therefore, cosine and secant are the even trigonometric functions, whereas sine, tangent, cosecant and cotangent are the odd trigonometric functions. If we know the even and odd trigonometric functions, it helps us to simplify the trigonometric expression when the variable inside the trigonometric function is negative.

## Trigonometry Angles

The trigonometry angles which are commonly used in trigonometry problems are 0 ° , 30 ° , 45 ° , 60 ° and 90 °. The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize. We will also show the table where all the ratios and their respective angle’s values are mentioned. To find these angles we have to draw a right-angled triangle, in which one of the acute angles will be the corresponding trigonometry angle. These angles will be defined with respect to the ratio associated with it.

For example, in a right-angled triangle,

Sin θ = Perpendicular/Hypotenuse

or θ = sin -1 (P/H)

θ = cos -1 (Base/Hypotenuse)

θ = tan -1 (Perpendicular/Base)

## Trigonometry Table

Check the table for common angles which are used to solve many trigonometric problems involving trigonometric ratios.

In the same way, we can find the trigonometric ratio values for angles beyond 90 degrees, such as 180°, 270° and 360°.

## Unit Circle

The concept of unit circle helps us to measure the angles of cos, sin and tan directly since the centre of the circle is located at the origin and radius is 1. Consider theta be an angle then,

Suppose the length of the perpendicular is y and of base is x. The length of the hypotenuse is equal to the radius of the unit circle, which is 1. Therefore, we can write the trigonometry ratios as;

## List of Trigonometry Formulas

The Trigonometric formulas or Identities are the equations which are true in the case of Right-Angled Triangles. Some of the special trigonometric identities are given below –

- Pythagorean Identities
- sin²θ + cos²θ = 1
- tan 2 θ + 1 = sec 2 θ
- cot 2 θ + 1 = cosec 2 θ
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ – sin²θ
- tan 2θ = 2 tan θ / (1 – tan²θ)
- cot 2θ = (cot²θ – 1) / 2 cot θ
- Sum and Difference identities-

For angles u and v, we have the following relationships:

- sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
- cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
- \(\begin{array}{l}tan(u+v) = \frac{tan(u)\ +\ tan(v)}{1-tan(u)\ tan(v)}\end{array} \)
- sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
- cos(u – v) = cos(u)cos(v) + sin(u)sin(v)
- \(\begin{array}{l}tan(u-v) = \frac{tan(u)\ -\ tan(v)}{1+tan(u)\ tan(v)}\end{array} \)
- If A, B and C are angles and a, b and c are the sides of a triangle, then,
- a/sinA = b/sinB = c/sinC

Cosine Laws

- c 2 = a 2 + b 2 – 2ab cos C
- a 2 = b 2 + c 2 – 2bc cos A
- b 2 = a 2 + c 2 – 2ac cos B

## Trigonometry Identities

The three important trigonometric identities are:

- tan² θ + 1 = sec² θ
- cot ² θ + 1 = cosec² θ

## Euler’s Formula for trigonometry

As per the euler’s formula,

e ix = cos x + i sin x

Where x is the angle and i is the imaginary number.

\(\begin{array}{l}\sin x=\frac{e^{i x}-e^{-i x}}{2 i}\\ \cos x=\frac{e^{i x}+e^{-i x}}{2}\\ \tan x=\frac{\left(e^{i x}-e^{-i x}\right)}{i\left(e^{i x}+e^{-i x}\right)}\end{array} \)

## Trigonometry Basics

The three basic functions in trigonometry are sine, cosine and tangent. Based on these three functions the other three functions that are cotangent, secant and cosecant are derived.

All the trigonometrical concepts are based on these functions. Hence, to understand trigonometry further we need to learn these functions and their respective formulas at first.

If θ is the angle in a right-angled triangle, then

Cos θ = Base/Hypotenuse

Tan θ = Perpendicular/Base

Perpendicular is the side opposite to the angle θ.

The base is the adjacent side to the angle θ.

The hypotenuse is the side opposite to the right angle

The other three functions i.e. cot, sec and cosec depend on tan, cos and sin respectively, such as:

Cot θ = 1/tan θ

Sec θ = 1/cos θ

Cosec θ = 1/sin θ

Cot θ = Base/Perpendicular

Sec θ = Hypotenuse/Base

Cosec θ = Hypotenuse/Perpendicular

## Trigonometry Examples

There are many real-life examples where trigonometry is used broadly.

If we have been given with height of the building and the angle formed when an object is seen from the top of the building, then the distance between object and bottom of the building can be determined by using the tangent function, such as tan of angle is equal to the ratio of the height of the building and the distance. Let us say the angle is ∝, then

Tan ∝ = Height/Distance between object & building

Distance = Height/Tan ∝

Let us assume that height is 20m and the angle formed is 45 degrees, then

Distance = 20/Tan 45°

Since, tan 45° = 1

So, Distance = 20 m

## Applications of Trigonometry

- Its applications are in various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, etc.
- It is also helpful to measure the height of the mountain, find the distance of long rivers, etc.

## Video Lesson on Applications of Trigonometry

## Trigonometry Problems and Solutions

Example 1 : Two friends, Rakesh and Vishal started climbing a pyramid-shaped hill. Rakesh climbs 315 m and finds that the angle of depression is 72.3 degrees from his starting point. How high is he from the ground?

Solution : Let m is the height above the ground.

To find: Value of m

To solve m, use the sine ratio.

Sin 72.3° = m/315

0.953 = m/315

m= 315 x 0.953

m=300.195 mtr

The man is 300.195 mtr above the ground.

Example 2: A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow?

Let x be the angle of elevation of the sun, then

tan x = 55/23 = 2.391

x = tan -1 (2.391)

or x = 67.30 degrees

## Trigonometry Questions

Practise these questions given here to get a deep knowledge of Trigonometry. Use the formulas and table given in this article wherever necessary.

Q.1: In △ABC, right-angled at B, AB=22 cm and BC=17 cm. Find:

(a) sin A Cos B

(b) tan A tan B

Q.2: If 12cot θ= 15, then find sec θ.

Q.3: In Δ PQR, right-angled at Q, PR + QR = 30 cm and PQ = 10 cm. Determine the values of sin P, cos P and tan P.

## Frequently Asked Questions on Trigonometry

What do you mean by trigonometry.

Trigonometry is one of the branches of mathematics which deals with the relationship between the sides of a triangle (right triangle) with its angles. There are 6 trigonometric functions for which the relation between sides and angles are defined. Learn more about trigonometry now by visiting BYJU’S.

## What are the six basic Trigonometric Functions?

There are 6 trigonometric functions which are:

- Sine function
- Cosine function
- Tan function
- Sec function
- Cot function
- Cosec function

## What is the formula for six trigonometry functions?

What is the primary function of trigonometry, who is the founder of trigonometry, what are the applications of trigonometry in real life.

One of the most important real-life applications of trigonometry is in the calculation of height and distance. Some of the sectors where the concepts of trigonometry are extensively used are aviation department, navigation, criminology, marine biology, etc. Learn more about the applications of trigonometry here.

Learn about Trigonometry in a simple manner with detailed information, along with step by step solutions to all questions, only at BYJU’S. Download the app to get personalised videos.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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## Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary. Follow the links for more, or go to Trigonometry Index

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

## Right-Angled Triangle

The triangle of most interest is the right-angled triangle . The right angle is shown by the little box in the corner:

Another angle is often labeled θ , and the three sides are then called:

- Adjacent : adjacent (next to) the angle θ
- Opposite : opposite the angle θ
- and the longest side is the Hypotenuse

## Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

## Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle " θ ":

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan .)

## Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = Opposite Hypotenuse = 2.8 4.9 = 0.57...

The triangle could be larger, smaller or turned around, but that angle will always have that ratio .

Calculators have sin, cos and tan to help us, so let's see how to use them:

## Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

- We know the Hypotenuse
- And we want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse :

sin(45°) = Opposite Hypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

What does the 0.7071... mean? It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse.

We can now put 0.7071... in place of sin(45°):

0.7071... = Opposite Hypotenuse

And we also know the hypotenuse is 20 :

0.7071... = Opposite 20

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Opposite = 14.14m (to 2 decimals)

The tree is 14.14m tall

## Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates , so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles !

## Unit Circle

What you just played with is the Unit Circle .

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

Note: you can see the nice graphs made by sine, cosine and tangent .

## Degrees and Radians

Angles can be in Degrees or Radians . Here are some examples:

## Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency ).

When we want to calculate the function for an angle larger than a full rotation of 360° (2 π radians) we subtract as many full rotations as needed to bring it back below 360° (2 π radians):

## Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

## Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2 π radians

−3 + 2 π = −3 + 6.283... = 3.283... rad ians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

## Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles . "Solving" means finding missing sides and angles.

## Example: Find the Missing Angle "C"

Angle C can be found using angles of a triangle add to 180° :

So C = 180° − 76° − 34° = 70°

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3 (except for the three angles case)

See Solving Triangles for more details.

## Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

## Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

Enjoy becoming a triangle (and circle) expert!

## What Is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles. Further still, trigonometry has astoundingly intricate relationships to other branches of mathematics, in particular complex numbers, infinite series, logarithms and calculus.

The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle ( trigōnon ) and measure ( metron ). Though the field emerged in Greece during the third century B.C., some of the most important contributions (such as the sine function) came from India in the fifth century A.D. Because early trigonometric works of Ancient Greece have been lost, it is not known whether Indian scholars developed trigonometry independently or after Greek influence. According to Victor Katz in “ A History of Mathematics (3rd Edition) ” (Pearson, 2008), trigonometry developed primarily from the needs of Greek and Indian astronomers.

## An example: Height of a sailboat mast

Suppose you need to know the height of a sailboat mast, but are unable to climb it to measure. If the mast is perpendicular to the deck and top of the mast is rigged to the deck, then the mast, deck and rigging rope form a right triangle. If we know how far the rope is rigged from the mast, and the slant at which the rope meets the deck, then all we need to determine the mast’s height is trigonometry.

For this demonstration, we need to examine a couple ways of describing “slant.” First is slope , which is a ratio that compares how many units a line increases vertically (its rise ) compared to how many units it increases horizontally (its run ). Slope is therefore calculated as rise divided by run. Suppose we measure the rigging point as 30 feet (9.1 meters) from the base of the mast (the run). By multiplying the run by the slope, we would get the rise — the mast height. Unfortunately, we don’t know the slope. We can, however, find the angle of the rigging rope, and use it to find the slope . An angle is some portion of a full circle, which is defined as having 360 degrees. This is easily measured with a protractor. Let’s suppose the angle between the rigging rope and the deck is 71/360 of a circle, or 71 degrees.

We want the slope, but all we have is the angle. What we need is a relationship that relates the two. This relationship is known as the “ tangent function,” written as tan(x). The tangent of an angle gives its slope. For our demo, the equation is: tan(71°) = 2.90. (We'll explain how we got that answer later.)

This means the slope of our rigging rope is 2.90. Since the rigging point is 30 feet from the base of the mast, the mast must be 2.90 × 30 feet, or 87 feet tall. (It works the same in the metric system: 2.90 x 9.1 meters = 26.4 meters.)

## Sine, cosine and tangent

Depending on what is known about various side lengths and angles of a right triangle, there are two other trigonometric functions that may be more useful: the “ sine function” written as sin(x), and the “ cosine function” written as cos(x). Before we explain those functions, some additional terminology is needed. Sides and angles that touch are described as adjacent . Every side has two adjacent angles. Sides and angles that don’t touch are described as opposite . For a right triangle, the side opposite to the right angle is called the hypotenuse (from Greek for “stretching under”). The two remaining sides are called legs .

Usually we are interested (as in the example above) in an angle other than the right angle. What we called “rise” in the above example is taken as length of the opposite leg to the angle of interest; likewise, the “run” is taken as the length of the adjacent leg. When applied to an angle measure, the three trigonometric functions produce the various combinations of ratios of side lengths.

In other words:

- The tangent of angle A = the length of the opposite side divided by the length of the adjacent side
- The sine of angle A = the length of the opposite side divided by the length of the hypotenuse
- The cosine of angle A = the length of the adjacent side divided by the length of the hypotenuse

From our ship-mast example before, the relationship between an angle and its tangent can be determined from its graph, shown below. The graphs of sine and cosine are included as well.

Worth mentioning, though beyond the scope of this article, is that these functions relate to each other through a great variety of intricate equations known as identities , equations that are always true.

Each trigonometric function also has an inverse that can be used to find an angle from a ratio of sides. The inverses of sin(x), cos(x), and tan(x), are arcsin(x), arccos(x) and arctan(x), respectively.

## Shapes other than right triangles

Trigonometry isn’t limited to just right triangles. It can be used with all triangles and all shapes with straight sides, which are treated as a collection of triangles. For any triangle, across the six measures of sides and angles, if at least three are known the other three can usually be determined. Of the six configurations of three known sides and angles, only two of these configurations can’t be used to determine everything about a triangle: three known angles ( AAA ), and a known angle adjacent and opposite to the known sides ( ASS ). Unknown side lengths and angles are determined using the following tools:

- The Law of Sines , which says that if both measures of one of the three opposing angle/side pairs are known, the others may be determined from just one known: sin(A)/a = sin(B)/b = sin(C)/c
- The Law of Cosines , which says that an unknown side can be found from two known sides and the angle between them. It’s essentially the Pythagorean Theorem with a correction factor for angles that aren’t 90 degrees: c2 = a2 + b2 – 2ab∙cos(C)
- The fact that all the angles in a triangle must add up to 180 degrees: A + B + C = 180°

## The history of trigonometry

Trigonometry follows a similar path as algebra : it was developed in the ancient Middle East and through trade and immigration moved to Greece, India, medieval Arabia and finally Europe (where consequently, colonialism made it the version most people are taught today). The timeline of trigonometric discovery is complicated by the fact that India and Arabia continued to excel in the study for centuries after the passing of knowledge across cultural borders. For example, Madhava’s 1400 discovery of the infinite series of sine was unknown to Europe up through Isaac Newton’s independent discovery in 1670. Due to these complications, we’ll focus exclusively on the discovery and passage of sine, cosine, and tangent.

Beginning in the Middle East, seventh-century B.C. scholars of Neo-Babylonia determined a technique for computing the rise times of fixed stars on the zodiac. It takes approximately 10 days for a different fixed star to rise just before dawn, and there are three fixed stars in each of the 12 zodiacal signs; 10 × 12 × 3 = 360. The number 360 is close enough to the 365.24 days in a year but far more convenient to work with. Nearly identical divisions are found in the texts of other ancient civilizations, such as Egypt and the Indus Valley . According to Uta Merzbach in “ A History of Mathematics ” (Wiley, 2011), the adaptation of this Babylonian technique by Greek scholar Hypsicles of Alexandria around 150 B.C. was likely the inspiration for Hipparchus of Nicea (190 to 120 B.C.) to begin the trend of cutting the circle into 360 degrees. Using geometry, Hipparchus determined trigonometric values (for a function no longer used) for increments of 7.5 degrees (a 48 th of a circle). Ptolemy of Alexandria (A.D. 90 to 168), in his A.D. 148 “ Almagest ”, furthered the work of Hipparchus by determining trigonometric values for increments of 0.5 degrees (a 720 th of a circle) from 0 to 180 degrees.

The oldest record of the sine function comes from fifth-century India in the work of Aryabhata (476 to 550). Verse 1.12 of the “ Aryabhatiya ” (499), instead of representing angles in degrees, contains a list of sequential differences of sines of twenty-fourths of a right angle (increments of 3.75 degrees). This was the launching point for much of trigonometry for centuries to come.

The next group of great scholars to inherit trigonometry were from the Golden Age of Islam. Al-Ma'mun (813 to 833), the seventh caliph of the Abbasid Caliphate and creator of the House of Wisdom in Baghdad, sponsored the translation of Ptolemy’s "Almagest" and Aryabhata’s "Aryabhatiya" into Arabic. Soon after, Al-Khwārizmī (780 to 850) produced accurate sine and cosine tables in “Zīj al-Sindhind” (820). It is through this work that that knowledge of trigonometry first came to Europe. According to Gerald Toomer in the “ Dictionary of Scientific Biography 7 ,” while the original Arabic version has been lost, it was edited around 1000 by al-Majriti of Al-Andalus (modern Spain), who likely added tables of tangents before Adelard of Bath (in South England) translated it into Latin in 1126.

Additional resources

- Math Is Fun: Trigonometry
- Khan Academy: Trigonometry
- Wolfram MathWorld: Trigonometry

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Robert Coolman, PhD, is a teacher and a freelance science writer and is based in Madison, Wisconsin. He has written for Vice, Discover, Nautilus, Live Science and The Daily Beast. Robert spent his doctorate turning sawdust into gasoline-range fuels and chemicals for materials, medicine, electronics and agriculture. He is made of chemicals.

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## Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

- Sine, Cosine and Tangent
- Pythagoras' Theorem
- The Sine and Cosine Rules
- Area of a Triangle
- Trigonometric Identities and Formulae
- Cosecant, Secant and Cotangent
- Hyperbolic Functions
- Euler's Formula and Euler's Identity

## IMAGES

## COMMENTS

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