Maths Trigonometry Chapter - Maths JEE Main

Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a right-angled triangle. It is one of the most important branches in the history of mathematics. Hipparchus, a Greek mathematician, introduced this concept. We will learn the fundamentals of trigonometry in this article, including trigonometry functions, ratios, trigonometry tables, formulas, and many solved examples.

JEE Main Maths Chapters 2024

Important Topics of Maths Trigonometry Chapter

• Trigonometric Ratios

• Trigonometry - Formulas, Identities, Functions and Problems

Maths Trigonometry Important Concept for JEE Main

What is Trigonometry?

The word Trigonometry is clubbed as, 'Trigonon' which means triangle and 'Metron' means to measure. The branch of mathematics known as "trigonometry" studies the relationship between the sides and angles of a right-angle triangle. Using trigonometric formulas, functions, or identities, it is possible to find the missing or unknown angles or sides of a right triangle. The angles in trigonometry can be measured in degrees or radians. 0°, 30°, 45°, 60°, and 90° are some of the most commonly used trigonometric angles in calculations.

Trigonometry is further divided into two subcategories. The following are the two types of trigonometry:

Plane Trigonometry and Spherical trigonometry

Basic Trigonometry

The measurement of angles and problems involving angles are covered in the fundamentals of trigonometry. Trigonometry has three basic functions: sine, cosine, and tangent. Other important trigonometric functions can be derived using these three basic ratios or functions: cotangent, secant, and cosecant. These functions are the foundation for all of the important concepts in trigonometry.

Ratios in Trigonometry: Sine, Cosine, and Tangent

The trigonometric functions are the trigonometric ratios of a triangle. The trigonometric functions sine, cosine, and tangent are abbreviated as sin, cos, and tan. Let's look at how these ratios or functions are evaluated in a right-angled triangle.

Consider a right-angled triangle with the longest side being the hypotenuse and the sides opposite to the hypotenuse being the adjacent and opposite sides.

Trigonometric Functions Formula

The primary trigonometric functions are sine, cosine, and tangent, represented as sin θ, cos θ, and tan θ. These functions describe the relationship between the angles of a right triangle and the ratios of its sides. Key properties of trigonometric functions include periodicity, amplitude, and phase shift.

If $\theta$ is the angle formed by the base and hypotenuse in a right-angled triangle, then

$\sin \theta = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}}$

$\cos⁡ \theta = \dfrac{\text{Base}}{\text{Hypotenuse}}$

$\tan⁡ \theta = \dfrac{\text{Perpendicular}}{\text{Base}}$

The values of the other three functions, cot, sec, and cosec, are determined by the values of tan, cos, and sin.

$\cot \theta = \dfrac{1}{\tan⁡ \theta} = \dfrac{\text{Base}}{\text{Perpendicular}}$

$\sec⁡ \theta = \dfrac{1}{\cos⁡ \theta} = \dfrac{\text{Hypotenuse}}{\text{Base}}$

$\text{cosec}{\theta} = \dfrac{1}{\sin \theta} = \dfrac{\text{Hypotenuse}}{\text{Perpendicular}}$

Even and Odd Trigonometric Functions

Even or odd can be used to describe the trigonometric function.

Odd Trigonometric Functions: If f(-x) = -f(x) and symmetric with respect to the origin, a trigonometric function is said to be odd.

Even Trigonometric Functions: If f(-x) = f(x) and symmetric to the y-axis, a trigonometric function is said to be even.

• $\sin⁡(-x) = -\sin⁡ \theta$

• $\cos⁡(-x) = \cos⁡ \theta$

• $\tan⁡(-x) = -\tan⁡ \theta$

• $\text{cosec}(-x) = -\text{cosec} \theta$

• $\sec⁡(-x) = \sec⁡ \theta$

• $\cot⁡(-x) = -\cot \theta$ 

Trigonometric Functions in Different Quadrants

Trigonometric Table - Trigonometry Table Formula

The trigonometric table is made up of interrelated trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent are used to calculate standard angle values. Refer to the below trigonometric table chart to know more about these ratios.

Similarly, we can find the trigonometric ratio values for angles other than 90 degrees, such as 180 degrees, 270 degrees, and 360 degrees.

Important Trigonometric Angles

Trigonometric angles are the angles in a right-angled triangle that can be used to represent various trigonometric functions. $0^\circ,30^\circ,45^\circ,60^\circ$ and $90^\circ$ are some of the standard angles used in trigonometry. These angles' trigonometric values can be found directly in a trigonometric table. $180^\circ,270^\circ$ and $360^\circ$ are some other important angles in trigonometry. The angle of trigonometry can be expressed in terms of trigonometric ratios as follows:

• $\theta = \sin^{-1} \dfrac{\text{Perpendicular}}{\text{Hypotenuse}}$

• $\theta = \cos⁡^{-1} \dfrac{\text{Base}}{\text{Hypotenuse}}$

• $\theta = \tan⁡^{-1} \dfrac{\text{Perpendicular}}{\text{Base}}$

Unit Circle

Because the center of the circle is at the origin and the radius is 1, the concept of unit circle allows us to directly measure the angles of cos, sin, and tan. Assume theta is an angle, and the length of the perpendicular is y and the length of the base is x. The hypotenuse is the same length as the radius of the unit circle, which is 1. As a result, the trigonometry ratios can be written as;

$\sin \theta = y$

$\cos \theta = x$

$-\tan \theta = \dfrac{y}{x}$

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. Pythagorean Identities

$\sin^2 \theta + \cos^2 \theta = 1$

$\tan^2 \theta + 1 = \sec^2 \theta$

$\cot^2 \theta + 1 = \text{cosec}^2 \theta$

$\sin 2 \theta = \sin^2 \theta \cos \theta$

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$

$\tan 2 \theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$

$\cot 2 \theta = \dfrac{\cot^2 \theta - 1}{2\cot \theta}$

2. Sine and Cosine Law in Trigonometry

Sine Law: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

Cosine Law: $c^2 = a^2 + b^2 - 2ab \cos C$

$a^2 = b^2 + c^2 - 2bc \cos A$

$b^2 = a^2 + c^2 - 2ac \cos B$

The lengths of the triangle's sides are a, b, and c, and the triangle's angle is A, B, and C.

3. Sum and Difference identities

Let u and v be the angles:

$\sin⁡(u+v) = \sin⁡(u)\cos⁡(v) + \cos⁡(u)\sin⁡(v)$

$\cos⁡(u+v) = \cos⁡(u)\cos⁡(v) - \sin⁡(u)sin⁡(v)$

$\tan⁡(u+v) = \dfrac{tan⁡(u)+\tan⁡(v)}{1 - \tan⁡(u)\tan⁡(v)}$

$\sin⁡(u-v) = \sin⁡(u)\cos⁡(v) - \cos⁡(u)\sin⁡(v)$

$\cos⁡(u-v) = \cos⁡(u)\cos⁡(v) + \sin⁡(u) \sin⁡(v)$

$\tan⁡(u-v) = \dfrac{\tan⁡(u) - \tan⁡(v)}{1+\tan⁡(u)\tan⁡(v)}$

4. Trigonometry Identities

$\sin^2 \theta + \cos^2 \theta = 1$

$\tan^2 \theta + 1 = \sec^2 \theta$

$\cot^2 \theta + 1 = \text{cosec}^2 \theta$

5. Euler's Formula for trigonometry

$e^{ix} = \cos x + i\sin x$

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

Hence Euler’s formula for sin,cos and tan is:

$\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$

$\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$

$\tan x = \dfrac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}$

Inverse Trigonometric Functions

Inverse trigonometric functions, denoted as $\sin^{(-1)}x, \cos^{(-1)}x$ , and $\tan^{(-1)}x$ , serve as the inverse operations of their respective trigonometric functions. They help find the angle when the trigonometric value is known. For example, if you know the sine of an angle, sin θ = x, you can use $\sin^{(-1)}$ to find the angle θ.

Properties of Trigonometric Functions

Trigonometric functions have various properties, such as even and odd functions, periodicity, and relations between functions. Some important properties of Inverse trigonometric function include:

• Even and Odd Functions: cos θ is an even function (symmetric about the y-axis), while sin θ is an odd function (symmetric about the origin).

• Periodicity: sin θ and cos θ have a period of 360° or 2π radians, while tan θ has a period of 180° or π radians.

• Addition and Subtraction Formulas: These formulas help express trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of those angles.

• Double Angle Formulas: These formulas allow you to express trigonometric functions of double angles in terms of trigonometric functions of the original angle.

Heights and Distances

Trigonometry's applications extend beyond the classroom and are particularly crucial in solving real-world problems. One such application is solving problems related to heights and distances, often encountered in fields like navigation, physics, and engineering. By using trigonometric principles, you can determine the height of objects, the distance between two points, or the angle of elevation or depression.

To solve height and distance problems, you typically use trigonometric ratios and create right triangles that model the situation. The key trigonometric functions, sin θ, cos θ, and tan θ, are essential in such calculations.

Trigonometry Formulas for JEE Main

Trigonometry is a crucial topic in mathematics, and a solid grasp of its formulas is essential for success in JEE Main. Here's a comprehensive guide to the key trigonometric formulas you need to master:

1. Basic Trigonometric Ratios:

$\text{Sine (}\sin\theta\text{)}$ &: $\frac{\text{Opposite}}{\text{Hypotenuse}}$

$\text{Cosine (}\cos\theta\text{)}$ &: $\frac{\text{Adjacent}}{\text{Hypotenuse}}$

$\text{Tangent (}\tan\theta\text{)}$ &: $\frac{\text{Opposite}}{\text{Adjacent}}$

2. Reciprocal Trigonometric Ratios:

$\text{Cosecant (}\csc\theta\text{)}$ &: $\frac{1}{\sin\theta}$

$\text{Secant (}\sec\theta\text{)}$ &: $\frac{1}{\cos\theta}$

$\text{Cotangent (}\cot\theta\text{)}$ &: $\frac{1}{\tan\theta}$

3. Pythagorean Identities:

$\sin^2\theta + \cos^2\theta$ & = 1

$1+\tan^2\theta$ & $= \sec^2\theta$

$1+\cot^2\theta$ & $= \csc^2\theta$

4. Sum and Difference Formulas:

$\sin(A \pm B)$ & $=\sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B)$ & $=\cos A \cos B \mp \sin A \sin B$

$\tan(A \pm B)$ & $=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

5. Double Angle Formulas:

$\sin(2\theta)$ & $= 2\sin\theta\cos\theta$

$\cos(2\theta)$ & $= \cos^2\theta - \sin^2\theta$

$\tan(2\theta)$ & $= \frac{2\tan\theta}{1 - \tan^2\theta}$

6. Half-Angle Formulas:

$\sin\left(\frac{\theta}{2}\right)$ & $= \pm\sqrt{\frac{1 - \cos\theta}{2}}$

$\cos\left(\frac{\theta}{2}\right)$ & $= \pm\sqrt{\frac{1 + \cos\theta}{2}}$

$\tan\left(\frac{\theta}{2}\right)$ & $= \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$

7. Product-to-Sum and Sum-to-Product Formulas:

$\sin A \sin B$ & $= 0.5[\cos(A - B) - \cos(A + B)]$

$\cos A \cos B$ & $= 0.5[\cos(A - B) + \cos(A + B)]$

8. Law of Sines and Cosines:

For any triangle ABC:

$[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}]$

$[a^2 = b^2 + c^2 - 2bc \cos A]$

9. Inverse Trigonometric Functions:

$(\sin^{-1}(x)), (\cos^{-1}(x)), (\tan^{-1}(x)), (\cot^{-1}(x)), (\sec^{-1}(x)), (\csc^{-1}(x))$

Understanding and applying these formulas will enhance your problem-solving skills and contribute to success in mastering trigonometry for the JEE Main exam. Regular practice and application in problem-solving are key to mastering these concepts effectively.

How All Trigonometric Formulas Help to Score Good In JEE Main?

In the context of JEE Main, a thorough understanding and application of all trigonometric formulas play a crucial role in solving complex mathematical problems. Here's how a command over these formulas contributes to scoring well:

• Problem Solving: Trigonometry is a fundamental part of mathematics and physics problems in JEE Main. Knowing Trigonometric formulas enables quick identification and application of the right trigonometric concepts to solve problems efficiently.

• Coordinate Geometry: Trigonometric formulas are often used in coordinate geometry. Understanding the relationships between angles and coordinates helps in solving geometric problems, especially in calculus and algebra.

• Physics Applications: Many physics problems involve concepts from trigonometry. From projectile motion to wave mechanics, a strong foundation in trigonometric formulas aids in tackling physics questions effectively.

• Calculus Integration: Trigonometric functions frequently appear in calculus, especially in integration problems. Familiarity with trigonometric identities and integrals is essential for solving integral calculus questions in the exam.

• Quick Problem Solving: In a time-bound exam like JEE Main, speed is crucial. Knowing all trigonometric formulas by heart allows students to quickly recognize patterns and apply the appropriate formulas, saving valuable time during the examination.

• Scoring in Mathematics Section: As a significant portion of the JEE Main mathematics section is dedicated to trigonometry, a good command over all trigonometric formulas ensures that students can secure a substantial number of marks in this section, contributing to an overall higher score.

Trigonometry Notes on Trigonometry Identities

• The trigonometry identities are the trigonometry equations that include all of the trigonometry ratios of all the angles

• Each trigonometric ratio can be expressed in terms of another trigonometric ratio.

• We can easily find the other value of the trigonometry ratio if we know one of the values of the trigonometry ratio.

• They can also be used to calculate trigonometric formulas.

Application of Trigonometry

The height of a structure or a mountain is calculated using trigonometry. The height of a building can be easily calculated using trigonometric functions and the distance of a structure from the perspective. It is used in a variety of fields and has no specific applications in solving functional problems. For example, in the development of computer music, trigonometry is used: as you may know, sound travels in waves, and this wave pattern is used in the development of computer music by passing it through a sine or cosine function.

JEE Main Maths Trigonometry Solved Examples 

Example 1:  A man stands in front of a 44 foot pole. According to his calculations, the pole cast a shadow that was 13 feet long. Can you assist him in determining the sun's angle of elevation from the shadow's tip?

Ans: Let x be the angle of elevation of the sun,

$\tan x = \dfrac{44}{13} = 3.384$

$x = \tan^{-1}⁡(3.384) = 1.283$

Hence, x in degree is $73.54^\circ$

Example 2: Find the value of $\sin ⁡75^\circ$

Ans: Given, $\sin ⁡75^\circ$

To find the value of $\sin ⁡75^\circ$ use the formula

$\sin⁡(A+B) = \sin⁡ A \cdot \cos B + \cos A \cdot \sin B$

Split $75^\circ$ such that $A = 30^\circ$ and $B = 45^\circ$

$\sin⁡ 75^\circ = \sin⁡(30^\circ + 45^\circ)$

$\Rightarrow \sin ⁡30^\circ \cdot \cos ⁡45^\circ + \cos ⁡30^\circ \cdot \sin ⁡45^\circ$

$\Rightarrow \dfrac{1}{2} \cdot \dfrac{1}{\sqrt{2}} + \dfrac{\sqrt{3}}{2} \cdot \dfrac{1}{\sqrt{2}}$

$\Rightarrow \dfrac{1}{2\sqrt{2}} + \dfrac{3}{2\sqrt{2}}$

$\Rightarrow \dfrac{\sqrt{3}+1}{2\sqrt{2}}$

Previous Year Questions from JEE Main Paper

1. Find the general solution of $\sin x − 3 \sin ⁡2x + \sin ⁡3x = \cos x - 3\cos ⁡2x + \cos ⁡3x$ is _________.

Ans: $\sin x − 3 \sin ⁡2x + \sin ⁡3x = \cos x - 3\cos ⁡2x + \cos ⁡3x$

$\Rightarrow 2\sin 2x \cos x - 3 \sin ⁡2x + \sin ⁡3x = \cos x - 3\cos ⁡2x + \cos ⁡3x$

$\Rightarrow 2 \sin ⁡2x \cos⁡ x - 3 \sin ⁡2x - 2\cos ⁡2x \cos x + 3\cos ⁡2x = 0$

$\Rightarrow \sin ⁡2x (2 \cos x - 3) - \cos ⁡2x (2 \cos x - 3)=0$

$\Rightarrow (\sin 2x - \cos ⁡2x)(2\cos x - 3)=0$

$\Rightarrow \sin ⁡2x = \cos⁡ 2x$

$\Rightarrow 2x = 2n\pi \pm \left(\dfrac{\pi}{2} - 2x\right)$

$x = \dfrac{n\pi}{2} + \dfrac{\pi}{8}$

2. Find the value of $\sin⁡(\cot^{-1} x)$?

Ans: Let $\cot^{-1} x = \theta$

Hence, $x = \cot \theta$

W.K.T $1 + \cot ⁡2\theta = \text{cosec}2\theta$

$\Rightarrow 1 + x^2 = \text{cosec} 2\theta$

W.K.T $\text{cosec} \theta = \dfrac{1}{\sin \theta}$

$\Rightarrow 1 + x^2 = \dfrac{1}{\sin^2 \theta}$�

$\Rightarrow \sin 2\theta = \dfrac{1}{1+x^2}$

$\Rightarrow \sin \theta = \sqrt{\dfrac{1}{1+x^2}}$

$\sin⁡(\cot^{-1}) = \dfrac{1}{\sqrt{1+x^2}}$

3. A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 meters and the distance between B and C is 100 meters, then the height of the balloon is given by _________.

Ans: 

$x = h \cot ⁡3\alpha$ —(i)

$(x+100) = h \cot ⁡2\alpha$  —(ii)

$(x+300) = h \cot \alpha$  —(iii)

From (i) and (ii), we get

$-100 = h (\cot ⁡3\alpha - \cot ⁡2\alpha) = h \dfrac{(\sin ⁡2\alpha \cos⁡3\alpha - \cos ⁡2\alpha \sin ⁡3\alpha)}{\sin ⁡3\alpha \sin ⁡2\alpha} = h \dfrac{\sin⁡(3\alpha - 2\alpha)}{\sin ⁡3\alpha \sin ⁡2\alpha}$

On simplifying we get,

$100 = h \left(\dfrac{\sin \alpha }{\sin ⁡3\alpha \sin⁡ 2\alpha}\right)$ —(iv)

Similarly,

From (ii) and (iii), we get

$-200 = h (\cot ⁡2\alpha - \cot \alpha) = h \dfrac{\sin⁡(2\alpha - \alpha)}{\sin⁡ 2\alpha \sin\alpha}$

On simplifying we get,

$200 = h \left(\dfrac{\sin \alpha}{\sin ⁡2\alpha \sin \alpha}\right)$ —(v)

Now divide equation (iv) and (v) we get,

$\dfrac{\sin ⁡3\alpha}{\sin \alpha} = \dfrac{200}{100}$

$\Rightarrow \dfrac{\sin ⁡3\alpha}{\sin \alpha} = 2$ —(vi)

W.K.T $\sin ⁡3\alpha = 3 \sin \alpha - 4 \sin^3\alpha$

So, From equation (vi) we get,

$\Rightarrow 3 \sin \alpha - 4 \sin^3\alpha - 2\sin \alpha=0$

$\Rightarrow 4 \sin^3 \alpha - \sin \alpha = 0 \Rightarrow \sin \alpha = 0$ or $\sin^2\alpha = \dfrac{1}{4}$

$\sin^2 \alpha = \dfrac{1}{4} = \sin^2 \dfrac{\pi}{6}$

$\Rightarrow \alpha = \dfrac{\pi}{6}$

Hence

$h = 200 \sin ⁡2\alpha = 200 \sin \dfrac{\pi}{3} = 200\dfrac{sqrt{3}}{2} = 100 \sqrt{3}$

So the height of the balloon is $100 \sqrt{3}$

Practice Problems

1. Find the value of $\sec^2⁡(tan^{-1}⁡2) + \text{cosec}^2(\cot^{-1}⁡3)$ = _________.

Ans: 15

2. If $\cos^{-1}p + \cos^{-1}q + \cos^{-1}r = \pi$ then $p^2 + q^2 + r^2 + 2pqr =$ ____________. $

Ans: 1

JEE Main Maths - Trigonometry Study Materials

Here, you'll find a comprehensive collection of study resources for Trigonometry designed to help you excel in your JEE Main preparation. These materials cover various topics, providing you with a range of valuable content to support your studies. Simply click on the links below to access the study materials of Trigonometry and enhance your preparation for this challenging exam.

JEE Main Maths Study and Practice Materials

Explore an array of resources in the JEE Main Maths Study and Practice Materials section. Our practice materials offer a wide variety of questions, comprehensive solutions, and a realistic test experience to elevate your preparation for the JEE Main exam. These tools are indispensable for self-assessment, boosting confidence, and refining problem-solving abilities, guaranteeing your readiness for the test. Explore the links below to enrich your Maths preparation.

Conclusion

Although trigonometry does not have many practical applications, it does make it easier to work with triangles. It's an excellent addition to geometry and actual measurements. With trigonometry, you can easily find the height without actually climbing a tree. They have a wide range of applications in real life and are extremely useful to most architects and astronomers. A standard trigonometry table helps solve subject-related problems. The 3 basic measures are sin, cos, and tan, and the remaining three are calculated using the formula given in the above list of formulas.