Trigonometry

Trigonometry connects angles and ratios. Mastery of identities is essential for JEE success.

Overview

graph TD
    A[Trigonometry] --> B[Basic Ratios]
    A --> C[Identities]
    A --> D[Equations]
    A --> E[Inverse Functions]
    C --> C1[Pythagorean]
    C --> C2[Sum/Difference]
    C --> C3[Multiple Angles]

Basic Trigonometric Ratios

In a right triangle:

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$

Reciprocal Ratios

$$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$$

Standard Values

Angle	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞

Fundamental Identities

Pythagorean Identities

$$\boxed{\sin^2\theta + \cos^2\theta = 1}$$ $$\boxed{1 + \tan^2\theta = \sec^2\theta}$$ $$\boxed{1 + \cot^2\theta = \csc^2\theta}$$

Quotient Identities

$$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

Sum and Difference Formulas

$$\boxed{\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B}$$ $$\boxed{\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B}$$ $$\boxed{\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}}$$

Double Angle Formulas

$$\boxed{\sin 2A = 2\sin A \cos A}$$ $$\boxed{\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A}$$ $$\boxed{\tan 2A = \frac{2\tan A}{1 - \tan^2 A}}$$

Half Angle Formulas

$$\sin^2\frac{A}{2} = \frac{1 - \cos A}{2}$$ $$\cos^2\frac{A}{2} = \frac{1 + \cos A}{2}$$ $$\tan\frac{A}{2} = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A}$$

Triple Angle Formulas

$$\sin 3A = 3\sin A - 4\sin^3 A$$ $$\cos 3A = 4\cos^3 A - 3\cos A$$ $$\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$$

Product-to-Sum Formulas

$$2\sin A \cos B = \sin(A+B) + \sin(A-B)$$ $$2\cos A \sin B = \sin(A+B) - \sin(A-B)$$ $$2\cos A \cos B = \cos(A+B) + \cos(A-B)$$ $$2\sin A \sin B = \cos(A-B) - \cos(A+B)$$

Sum-to-Product Formulas

$$\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}$$ $$\sin C - \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2}$$ $$\cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}$$ $$\cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2}$$

General Solutions

sin θ = sin α

$$\theta = n\pi + (-1)^n\alpha$$

cos θ = cos α

$$\theta = 2n\pi \pm \alpha$$

tan θ = tan α

$$\theta = n\pi + \alpha$$

Inverse Trigonometric Functions

Principal Values

Function	Domain	Range
sin⁻¹x	[-1, 1]	[-π/2, π/2]
cos⁻¹x	[-1, 1]	[0, π]
tan⁻¹x	ℝ	(-π/2, π/2)
cot⁻¹x	ℝ	(0, π)
sec⁻¹x	(-∞,-1]∪[1,∞)	[0,π]-{π/2}
csc⁻¹x	(-∞,-1]∪[1,∞)	[-π/2,π/2]-{0}

Important Properties

$$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$$ $$\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$$ $$\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$$ $$\sin^{-1}(-x) = -\sin^{-1}x$$ $$\cos^{-1}(-x) = \pi - \cos^{-1}x$$ $$\tan^{-1}(-x) = -\tan^{-1}x$$

Addition Formulas

$$\tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\frac{x+y}{1-xy} & xy < 1 \\ \pi + \tan^{-1}\frac{x+y}{1-xy} & xy > 1, x,y > 0 \\ -\pi + \tan^{-1}\frac{x+y}{1-xy} & xy > 1, x,y < 0 \end{cases}$$

Conversion Formulas

$$\sin^{-1}x = \tan^{-1}\frac{x}{\sqrt{1-x^2}}$$ $$\cos^{-1}x = \tan^{-1}\frac{\sqrt{1-x^2}}{x}$$

Practice Problems

1. Prove: $\tan 20° \tan 40° \tan 60° \tan 80° = 3$

2. Solve: $\sin x + \cos x = \sqrt{2}$

3. Find: $\tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3)$

4. Simplify: $\cos^{-1}(\frac{4}{5}) + \tan^{-1}(\frac{3}{5})$

Further Reading

• Complex Numbers
• Coordinate Geometry