Trigonometry

Introduction

Trigonometry is a crucial topic in the JEE Main Mathematics syllabus. It involves the study of relationships between the angles and sides of triangles, particularly right-angled triangles. This document will break down complex ideas into smaller, digestible sections, covering all necessary concepts in detail.

Trigonometric Ratios

Basic Trigonometric Ratios

The primary trigonometric ratios are sine, cosine, and tangent, defined for a right-angled triangle as follows:

• Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$

Example

For a right-angled triangle with an angle $\theta = 30^\circ$, the sides are:

• Opposite = 1

• Adjacent = $\sqrt{3}$

• Hypotenuse = 2

Then, $$ \sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \tan(30^\circ) = \frac{1}{\sqrt{3}} $$

Reciprocal Trigonometric Ratios

The reciprocal trigonometric ratios are cosecant, secant, and cotangent:

• Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
• Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
• Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$

Tip

Remember the mnemonic: "SOH-CAH-TOA" for sine, cosine, and tangent, and "CHOSHACOT" for their reciprocals.

Trigonometric Identities

Pythagorean Identities

The fundamental Pythagorean identities are derived from the Pythagorean theorem:

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

Angle Sum and Difference Identities

For any angles $\alpha$ and $\beta$:

• $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
• $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
• $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

Example

Given $\alpha = 45^\circ$ and $\beta = 30^\circ$,

$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Double Angle Identities

For any angle $\theta$:

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

Note

Double angle identities are especially useful in simplifying complex trigonometric expressions.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle given a trigonometric ratio. They are denoted as $\sin^{-1}(x)$, $\cos^{-1}(x)$, $\tan^{-1}(x)$, etc.

Principal Values

The principal values of inverse trigonometric functions are:

• $\sin^{-1}(x)$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$
• $\cos^{-1}(x)$: $[0, \pi]$
• $\tan^{-1}(x)$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

Common Mistake

A common mistake is to assume that $\sin^{-1}(x)$ and $\frac{1}{\sin(x)}$ are the same. They are not; $\sin^{-1}(x)$ is the inverse function, while $\frac{1}{\sin(x)}$ is the reciprocal.

Example

If $\sin(\theta) = \frac{1}{2}$, then $\theta = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ$ or $\frac{\pi}{6}$ radians.

Graphs of Trigonometric Functions

Sine and Cosine Functions

The graphs of $\sin(x)$ and $\cos(x)$ are periodic with a period of $2\pi$.

• Sine Function: Starts at (0,0), reaches its maximum at $\frac{\pi}{2}$, crosses zero at $\pi$, reaches its minimum at $\frac{3\pi}{2}$, and completes a cycle at $2\pi$.
• Cosine Function: Starts at (0,1), crosses zero at $\frac{\pi}{2}$, reaches its minimum at $\pi$, crosses zero again at $\frac{3\pi}{2}$, and completes a cycle at $2\pi$.

Tangent Function

The graph of $\tan(x)$ has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer, and is periodic with a period of $\pi$.

Tip

Understanding the graphs of trigonometric functions can help in visualizing and solving equations and inequalities involving these functions.

Trigonometric Equations and Inequalities

Solving Basic Trigonometric Equations

To solve equations like $\sin(x) = a$, $\cos(x) = b$, or $\tan(x) = c$, use the corresponding inverse functions and consider the periodic nature of trigonometric functions.

Example

Solve $\sin(x) = \frac{1}{2}$.

$$ x = \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x = \pi - \frac{\pi}{6} + 2n\pi = \frac{5\pi}{6} + 2n\pi \quad \text{for integer } n $$

Trigonometric Inequalities

To solve inequalities like $\sin(x) > a$ or $\cos(x)

< b$, use the unit circle and the properties of trigonometric functions.

Example

Solve $\sin(x) > \frac{1}{2}$.

$$ x \in \left(\frac{\pi}{6} + 2n\pi, \frac{5\pi}{6} + 2n\pi \right) \quad \text{for integer } n $$

Conclusion

Trigonometry is a vast and essential topic in the JEE Main Mathematics syllabus. Understanding the fundamental ratios, identities, and functions, as well as their applications, is crucial for solving complex problems. Practice regularly, and use this study guide to reinforce your understanding and improve your problem-solving skills.