Trigonometry
2026-02-28 12:49 Diff
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What is Trigonometry?

The word trigonometry is derived from the Greek words trigonon, meaning triangle, and metron, meaning measure. The principles of trigonometry include the measurement of angles and problems involving angles. This can be best understood with the help of a right triangle.

Three basic functions of trigonometry are sine, cosine, and tangent. These functions can be used to build other critical trigonometric functions such as cotangent, secant, and cosecant. It is important to know the three sides of a right-angled triangle to understand trigonometry. The three sides are:

  • Perpendicular (P): The side that is opposite to the angle ‘A’.
     
  • Base (B): The neighboring side of angle ‘A’.
     
  • Hypotenuse (H): It is the side opposite to the right angle in a right-angled triangle.


The trigonometric functions are determined by using specific formulas. The trigonometric formulas are as follows:

  • \(\text {sin A} = {P \over H }\)
     
  • \(\text {cos A} = {B \over H}\)
     
  • \(\text{tan A} = {P\over B}\)
     
  • \(\text{cosec A} = {H \over P}\)
     
  • \(\text{sec A} = {H \over B}\)
     
  • \(\text{cot A} = {B \over P}\)


Here, P, B, and H are perpendicular, base, and height of the right-angled triangle respectively.
 

Law of sines: The law of sines is a fundamental formula in trigonometry that relates the sides and angles of any triangle (not just right triangles).

For a triangle 𝐴𝐵𝐶 with sides 𝑎, 𝑏, 𝑐 opposite to angles 𝐴, 𝐵, 𝐶 the law of sines can be given as:

\(\frac{a}{sin \ A}​=\frac{b}{sin\ B}​=\frac{c}{sin\ C}​=2R\)

Where, R is the radius of the circumcircle of the triangle.

Law of cosines: The Law of Cosines is a key formula in trigonometry. It generalizes the Pythagorean theorem to non-right triangles.

For a triangle ABC with sides a, b, c opposite to angles A, B, C:

The law of cosines formula can be given as:

\(a^2=b^2+c^2-2bc\ cos\ A\\[1em] b^2=a^2+c^2-2ac\ cos\ B\\[1em] c^2=a^2+b^2-2ab\ cos\ C\)

History of Trigonometry

The ancient Egyptians and Babylonians first developed trigonometry to help with practical tasks like building structures and studying the stars. However, Greek mathematician Hipparchus developed it as a mathematical discipline in the 2nd century BCE, creating the first trigonometric tables using chords.
 

  • Ancient Egyptians employed early forms of trigonometry to construct their pyramids.
     
  • The Greeks shared an equal interest in triangles. Hipparchus, the father of trigonometry, created the first trigonometric tables.
     
  • The Greeks shared an equal interest in triangles. Hipparchus, the father of trigonometry, created the first trigonometric tables.
     
  • Indian mathematicians, such as Aryabhata, developed advanced concepts regarding angles.
     
  • Furthermore, Islamic scholars developed additional ideas, such as using trigonometry to navigate across deserts and seas.

Concepts of Trigonometry

The concepts of trigonometry refer to the important ideas that the subject covers. Let’s take a look at these concepts one by one.

  • Angles: Measured in degrees or radians.
     
  • Trigonometric ratios: Sine, cosine, tangent, cotangent, secant, and cosecant.
     
  • Heights and distances: Trigonometry and its real-life applications.
     
  • Graphs of trigonometric functions: Helps us understand how sine, cosine, and tangent behave.
     
  • Trigonometric identities: These are basically formulas like \(​sin^2 \theta + cos^2 \theta = 1 ​\).
     
  • Pythagoras’ theorem: It gives us the relation between the three sides of a right-angled triangle.
     
  • Right-angled triangles: This is the foundation of trigonometric ratios.  

Let's now understand the three sides of a right-angled triangle.

  • Hypotenuse: It is the longest side/arm of the triangle.
     
  • Opposite side: The side opposite to the angle that we are supposed to find is called the opposite side.
     
  • Adjacent side: The side next to the angle you are focusing on, but it is not the hypotenuse.

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Trigonometric Ratios

The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

Let's meet the three most important ratios:

1. Sine (sin) - The height finder 

Sine of an angle is defined by the ratio of length of sides which is opposite to the angle and the hypotenuse.

\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

Inverse of sine: The inverse of sine is known as the arcsine function. 

If \(y=sin(x)\),

Then the inverse of this function is, 

\(x = sin^{-1}(y) \ or \ x = arcsin(y)\)

2. Cosine (cos) - The adjacent to the hypotenuse.

The cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. 

 \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
​
Inverse of cos: The inverse of cos is known as the arccos function. 

If \(y=cos(x)\),

Then the inverse of this function is, 

\(x = cos^{-1}(y) \ or \ x = arccos(y)\)

​
3. Tangent (tan) - The angle decider

Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle.

\(​ tan \space\theta = {opposite \over adjacent }\)

\(​ tan \space\theta = {sin \space \theta\over cos \space \theta}\)

Trigonometric Functions

Trigonometric functions are six basic functions where the input is the angle of a right triangle, and the output is a numerical value.

1. Cosecant (Cosec) - The reverse of sine 

It is the reciprocal of sine function. It is defined as the ratio of hypotenuse to the opposite side.

\( cosec(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}} \)

\(cosec(\theta) = \frac{1}{\sin(\theta)} \)

2. Secant (sec) - The reverse of cosine

Secant is the reciprocal of cosine function. It gives the ratio of hypotenuse to the adjacent side.

\( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}}\)

\(\sec(\theta) = \frac{1}{\cos(\theta)} \)

3. Cotangent (cot) - The reverse of tangent

Cotangent is the reciprocal of tangent function. It is how we compare the adjacent side to the opposite side.

\( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \)

\(\cot(\theta) = \frac{1}{\tan(\theta)} \)

The following table represents the trigonometric functions and its relations to the ratio of sides:

Functions Abbreviations Relationship to sides of a right triangle Sine function Sin \(\frac{\text{Opposite side}}{\text{Hypotaneuse}}\) Cosine function Cos \(\frac{\text{Adjacent side}}{\text{Hypotaneuse}}\) Tangent function Tan \(\frac{\text{Opposite side}}{\text{Adjacent side}}\) Cosecant function Cosec \(\frac{\text{Hypotaneuse}}{\text{Opposite side}}\) Secant function Sec \(\frac{\text{Hypotaneuse}}{\text{Adjacent side}}\) Cotangent function Cot \(\frac{\text{Adjacent side}}{\text{Opposite side}}\)

Even and Odd Trigonometric Functions

The trigonometric functions are classified into even and odd function on the basis of their symmetrical relations. Let's understand each functions individually.

Odd functions

In trigonometry, a function f(x) is said to be odd if it satisfies the following property: f(−x) = −f(x). This means that flipping the input will also cause a flip in the output.

The following trigonometric functions are odd function:

  • \(sin(-x)= -sin(x)\)

    The sine function flips its sign when the angle is negative, as it is odd.

  • \(tan(-x) = \frac{sin(-x)}{cos(-x)} = -tan(x)\)

    As we know, tan x is defined as \(sin(x) \over cos(x)\). Since, the sine function is negative, and the cos function remains the same, the negative sine function affects the tangent. Hence, the tan function is also negative.

  • \(cosec(-x)= -cosec(x)\)

    Here, the cosecant is the reciprocal of sine. As the sine function changes sign, the cosecant function also changes sign.

  • \(cot(-x)= -cot(x)\)


    Since, cot is the reciprocal of tan function, its sign varies corresponding to the sign of tan.

Even functions

A function f(x) is said to be an even function if it satisfies the property: f(−x) = f(x). Here, flipping the input doesn’t change the output.

The even trigonometric functions are given below:

  • \(cos(-x)= cos(x)\)

    In the cosine function, the value of x remains positive for negative angles.

  • \(sec(-x)= sec(x)\)

    The secant function is the reciprocal of cosine, hence it is positive.

Trigonometry Ratios for Different Angles

Trigonometric ratios can be applied to common angles. The table given below shows the value of trigonometric ratios at certain angles. 

First, let's learn about how to create a trigonometric ratio table.

Step 1: Write the first five whole numbers 

Let’s start by writing the first five whole numbers. Imagine we are counting in a game of hopscotch, with some spaces in between each number so we can move to the next position.

\(0 \space |\space 1\space | \space2\space | \space3\space |\space 4\)

Step 2: Divide each number by 4

Now, we divide each number by 4. Here, the number is divided by 4 to make it easier for students to understand. 


\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{2}{4} \;\big|\; \tfrac{3}{4} \;\big|\; \tfrac{4}{4} \)
 

When we simplify \({2 \over 4}\) and \({4 \over 4}\), we write it as:

\( 0 \;\big|\; \tfrac{1}{4} \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{3}{4} \;\big|\; 1 \)

Step 3: Take the square root of each resulting number

\( 0 \;\big|\; \sqrt{\tfrac{1}{4}} \;\big|\; \sqrt{\tfrac{1}{2}} \;\big|\; \sqrt{\tfrac{3}{4}} \;\big|\; 1 \)

Step 4: Sine values for angles 0°, 30°, 45°, 60°, and 90°

Simplify each values:

\( \sin(\theta) : \; 0 \;\big|\; \tfrac{1}{2} \;\big|\; \tfrac{1}{\sqrt{2}} \;\big|\; \tfrac{\sqrt{3}}{2} \;\big|\; 1 \)

Step 5: Reverse the order for cosine values

Now, let's have some fun and reverse the order of sine values to find the cosine values. 

\( \: \; 1 \;\big|\; \frac{\sqrt{3}}{2} \;\big|\; \frac{1}{\sqrt{2}} \;\big|\; \frac{1}{2} \;\big|\; 0 \)

Step 6: Find the tangent values

Now we find the tangent values, which are simply the sine that is divided by the cosine.

  • \( \tan 0^\circ = {0\over 1} = 0\)
     
  • \( \tan 30^\circ = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \)
     
  • \( \tan 45^\circ = \frac{1/\sqrt{2}}{1/\sqrt{2}} = 1 \)
     
  • \( \tan 60^\circ = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \)
     
  • \( \tan 90^\circ = 1/0 = \text {Not defined.}\)


The following chart represents the values of sine, cosine and tangent function for basic angles in a tabular form.
 

Trigonometric ratios(\(\theta\)) \(0^{\circ} \) \(30^{\circ} \) \(45^{\circ}\) \(60^{\circ}\) \(90^{\circ}\) \(\text{Sin} \theta\) 0 \(\frac{1}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{\sqrt{3}}{2}\) 1 \(\text{cos}\theta\) 1 \(\frac{\sqrt{3}}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{1}{2}\) 0 \(\text{tan} \theta\) 0 \(\frac{1}{\sqrt{3}}\) 1 \(\sqrt{3}\) undefined \(\text{cosec}\theta\) undefined 2 \(\sqrt{2}\) \(\frac{2}{\sqrt{3}}\) 1 \(\text{sec}\theta\) 1 \(\frac{2}{\sqrt{3}}\) \(\sqrt{2}\) 2 undefined \(\text{cot}\theta\) undefined \(\sqrt{3}\) 1 \(\frac{1}{\sqrt{3}}\) 0

Trigonometry Formulas

Trigonometry formulas are used to solve trigonometric problems. These problems may include trigonometric ratios like sin, cos, tan, sec, cosec, and cot. Memorizing these mathematic formulas in trigonometry will help students solve problems easily. Here are some of the commonly used trigonometry formulas:

1. Pythagoras trigonometric identities 
 

  • \(\sin^2 \theta + \cos^2 \theta = 1\)
     
  • \(1 + \tan^2 \theta = \sec^2 \theta\)
     
  • \(csc^2 \theta = 1 + \cot^2 \theta \)

2. Double angle identities

  • \(sin(2\theta) = 2 \sin \theta \cos \theta \)
     
  • \(cos(2\theta) = \cos^2 \theta - \sin^2 \theta \)
     
  • \(tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)
     

3. Product - Sum identities

  • \(\sin A + \sin B = 2 \, \sin\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right)\)
     
  • \(\sin A - \sin B = 2 \, \cos\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right) \)
     
  • \(\cos A + \cos B = 2 \, \cos\left(\frac{A+B}{2}\right) \, \cos\left(\frac{A-B}{2}\right)\)
     
  • \(cos A - \cos B = -2 \, \sin\left(\frac{A+B}{2}\right) \, \sin\left(\frac{A-B}{2}\right)\)

4. Sum and difference identities

  • \( \sin(A+B) = \sin A \cos B + \cos A \sin B\)
     
  • \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)
     
  • \(\\cos(A+B) = \cos A \cos B - \sin A \sin B\)
     
  • \(\\cos(A-B) = \cos A \cos B + \sin A \sin B\)
     
  • \(\\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
     
  • \(\\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)
     

5. Trick formulas

Remember these ratios as a trick formula to easily access through problems.

\(sinθ=\frac{H}{O}​,\ cosθ=\frac{H}{A}​,\ tanθ=\frac{A}{O}​\)

6. Degrees to radiance

Trigonometric functions in calculus and advanced math are defined using radians, not degrees.

The conversion formula for degree to radiance is given as:

\(Radiance = Degrees ×\frac{π}{180​}\)

The radiance to degrees formula is given as:

\(Degrees =Radiance ×\frac{180}{π​}\)

8. Reciprocal identities

Reciprocal identities show how the six trigonometric functions are reciprocals (inverses in multiplication) of each other.

The foundation of many trigonometric simplifications and proofs are formed by the Reciprocal identities.

The reciprocal identities of the functions are given as;

\(sin\ θ=\frac{1}{csc\ θ}\)

\(cos\ θ=\frac{1}{sec\ θ}\)

\(tan\ θ=\frac{1}{cot\ θ}\)

\(csc\ θ=\frac{1}{sin\ θ}\)

\(sec\ θ=\frac{1}{cos\ θ}\)

\(cot\ θ=\frac{1}{tan\ θ}\)

Tips and Tricks to Remember Trigonometry

Trigonometry is all about angles and sides of a right-angled triangle. Kids can easily get confused by it. To master trigonometry, you can follow certain tips and tricks. In this section, let’s learn a few trigonometry tips and tricks.

1. SOH-CAH-TOA helps you remember the basics of sine, cosine, and tangent ratios:

  • SOH: \(sine = {opposite \over {hypotenuse}}\)
     
  • CAH: \(cosine = \frac {adjacent} {hypotenuse}\)
     
  • TOA: \(tangent = \frac {opposite} {adjacent}\)

2. You can also create a fun sentence/phrase to memorize, like “Silly Owls Help Cats And Turtles”.

3. You can also make a trigonometric ratio triangle.

4. Draw a triangle and label its sides as opposite, adjacent, and hypotenuse to avoid confusion between sides.

5. Use color coding for different ratios and functions.

Common Mistakes and How to Avoid Them in Trigonometry

Hey kids! Trigonometry can be a little tricky sometimes, and we tend to make mistakes. Don't worry, it happens to everyone. Let's get to know some common mistakes that students often make in trigonometry and how to avoid them.

Real-life Applications of Trigonometry

Trigonometry is a field of math that deals with the relationship between angles and sides of triangles. It has many practical applications, from measuring heights and distances to understanding sound and light waves. Trigonometry is applied in various fields like physics, engineering, computer science, etc.

Here are some fun and relatable real-life applications of trigonometry:

  1. Precision: Architects apply trigonometry to design buildings and structures, ensuring perfect angles and measurements.
     
  2. Navigating Air and Sea: Pilots and sailors use trigonometry to navigate safely.
     
  3. Engineering with Precision: Engineers rely on trigonometry to design machines and stable structures.
     
  4. Beyond Earth and Into Sports: Trigonometry plays a vital role in both astronomy and sports, enabling accurate calculations and analysis.
     
  5. Computer Graphics: Trigonometry is used to model movements of objects in 2-D plane. They are used for tracking motion in animations, graphic and video games.

1. Designing with Precision:

1. Designing with Precision:

Architects use trigonometry to design buildings and structures. They calculate angles and lengths to ensure everything fits together properly.

2. Navigating the Skies and Seas:

2. Navigating the Skies and Seas:

Pilots and sailors rely on trigonometry for navigation.

3. Engineering with Accuracy:

3. Engineering with Accuracy:

Trigonometry is important for engineers to design machines and structures.

4. Exploring the Universe and Sports:

4. Exploring the Universe and Sports:

Trigonometric calculations are also used in astronomy and sports.

5. Computer Graphics

5. Computer Graphics

Rotation of objects, movements of characters are programmed using trigonometry.

Problem 1

Find Sin 30° + cos 60°.

Okay, lets begin

\(\sin30^∘+\cos60^∘=1\)

Explanation

We know that,

\(\sin 30° = \frac {1}{2} \\[1em] \cos 60° = \frac {1}{2}\)

  1. Add them:

    \(\sin 30° + \cos 60° \)

  2. Simplify the values:

    \(\frac {1}{2} + \frac {1}{2} = 1\)

  3. Substitute the values:

    \(\sin30^∘+\cos60^∘=1\)

Well explained 👍

Problem 2

If tan θ = 1, find θ in degrees.

Okay, lets begin

\(θ = 45°\)

Explanation

We know that,

\(\tan θ = 1\)

So, \(\tan θ = 1\)

Therefore,
\(θ = 45°\)

Well explained 👍

Problem 3

Prove that sin² 45° + cos² 45° = 1

Okay, lets begin

\(\sin^245^∘+\cos^245^∘=1\)

Explanation

We know that,
 

\(\sin 45° = \frac {1}{\sqrt2}\\[1em] \cos 45° = \frac {1}{\sqrt2}\)

  1. By using the Pythagorean identity.

    \(\sin^2 45° + \cos^2 45°\)

  2. Substituting the values,

    \(= \bigg (\frac {1}{\sqrt2 } \bigg)^2 + \bigg (\frac {1}{\sqrt2 } \bigg)^2 \\\space \\ = \frac {1}{2 } + \frac {1}{2 } \\\space\\ = {2 \over 2} \\\space \\= 1\)


\(sin^245^∘+cos^245^∘=1\)

Well explained 👍

Problem 4

If sin A = 5/13, find sec A.

Okay, lets begin

\( \cos A = \frac{12}{13}, \quad \sec A = \frac{13}{12} \)

Explanation

Given:

\(\sin A = {5 \over 13}\)

We know that,

\(\sin^2 A + \cos^2 A = 1\)

  1. Substituting values:

    \(\cos^2A = 1 \space  – \frac {5}{13}^2 \\ \space \\ = 1 \space – \frac {25}{169}\)

  2. Converting 1 to have the same denominator,

    \(1 = \frac{169}{169}\)

  3. So the equation becomes,

    \(= \frac{169}{169} - \frac{25}{169} \\ \space\\ = \frac{144}{169}\)

    \(\cos A = \frac{2}{13}\)

  4. Taking reciprocal:

    \(​​\)\( \sec A = \frac {1}{\cos A} = \frac{1}{\frac{12}{13}}, \\ \space \\ \sec A = \frac{13}{12} \)

Well explained 👍

Problem 5

Prove that 1 + tan²θ = sec² θ

Okay, lets begin

\(1+tan^2θ=sec^2θ\)

Explanation

 We know that,

\(\tan θ = {\sin θ \over \cos θ}\)

  1. Take the square of tan θ 
    \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)
     
  2. Add 1\(1 + \tan^2 θ = 1 + \frac {\sin^2 θ} { \cos^2 θ}\)

  3. Write with a common denominator
    \( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \)
     

  4. Now, use Pythagorean identity:

    \( \sin^2 \theta + \cos^2 \theta = 1 \)

    \( \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \)


Therefore, we get \(1+tan^2θ=sec^2θ\)

Well explained 👍

FAQs on Trigonometry

1.What are the six trigonometric ratios of trigonometry that my child need to know?

The six trigonometric ratios your child need to know are sine, cosine, tangent, cotangent, cosecant, and secant.
 

2.How can I encourage my child to practice trigonometry?

Encourage them to use trigonometry in the real world. For example, finding the height of a tree, or finding the slope of your driveway, etc
 

3.How can by child easily memorize values of trigonometric function?

Use mnemonics like SOH, CAH and TOA to explain the formulas of trigonometric ratios to your child.

4.How can my child find the hypotenuse?

To find the hypotenuse, teach your child the Pythagoras theorem. According to Pythagoras theorem, the square of the hypotenuse is equal to the sum of the square of the other two sides.
 

5.How to explain trigonometry to my child?

Take a real life triangular object. Explain the sides and the ratios with respect to the angle. The visualization will help your child to grasp the concept easily.

6.Is it required for my child to memorize all trigonometric identites?

Yes, the identities are important for children to remember. Using identities simplifies complex problems. 

7.Will my child ever use trigonometry in real life?

Yes, your child will use trigonometry in real-life situations. Fields like engineering, astronomy, physics, and computer graphics generally uses concepts of trigonometry.

8.Is it important for my child to learn trigonometry if he/she is interested in medicine?

Yes, trigonometry is used in medicine for medical imaging, such as CT and MRI scans. These scans help reconstruct 3D images from 2D data, enabling doctors to understand the electrical axis and rhythm of the heart. It is also used in ultrasound to calculate the trajectories of needles. Studying trigonometry will help him/her to better understand how machines works.

9.How can I help my child to learn trigonometry?

Use triangular objects to help your child visualize the sides and the angles of the triangle. Encourage practicing writing formulas.

10.My child dreams of being a chemist. Is trigonometry important then?

Yes. To calculate bond lengths and bond angles within molecules, model their 3D shapes, and find molecular properties, trigonometric concepts are used by chemist. Studying trigonometry will help your child to understand complex problems easily in the future.

11.Is it necessary to learn trigonometry if my child want to be a pilot?

Yes. Pilots use trigonometry to calculate flight paths, descent angles, and wind correction angles. These calculations are based on triangles formed by the aircraft’s position, altitude, distance, and wind vector.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.