Chapter 3: Trigonometric Functions

In Class X you met $\sin$, $\cos$ and $\tan$ as ratios in a right triangle , defined only for acute angles. The Class-XI viewpoint is bigger: trigonometric functions become functions of any real number, defined by the position of a point on the unit circle. With this single change, $\sin x$ has meaning for $x = 1000$ or $x = -\pi/7$, and we can talk about its graph, its period, and its derivatives , in the next two years' worth of mathematics.

The shift starts with radian measure. A radian is the natural mathematical unit of angle: the length of the arc cut off on a unit circle. Once you measure angles in radians, formulas lose their stray $\pi/180$ factors and the calculus of trigonometric functions becomes clean.

You then learn the identities , equations between trigonometric expressions that hold for every angle. The Pythagorean identities, the sum and difference formulas $\sin(A \pm B), \cos(A \pm B)$, the multiple-angle formulas $\sin 2A, \cos 2A$, and the product-to-sum formulas. These dozen formulas, properly memorised, are the toolkit you will reach for in every problem in trigonometry, vectors, complex numbers and calculus.

We close with trigonometric equations , equations like $\sin x = 1/2$ , and their general solutions. Because trigonometric functions are periodic, every such equation has infinitely many solutions, and you must learn to write all of them in one expression.

For JEE and Class XII, this chapter is the foundation for: complex numbers in polar form, inverse trigonometric functions, calculus of trigonometric functions, vectors, parametric equations of curves, and differential equations. It pays to overlearn the identities now.

What's inside

1. Angles and radian measure , degrees, radians, conversion, arc length.
2. Trigonometric functions of any angle , unit circle definition, signs in quadrants.
3. Domain, range, and graphs , of $\sin, \cos, \tan, \cot, \sec, \csc$.
4. Trigonometric identities , Pythagorean, reciprocal, quotient, and key transformations.
5. Sum, difference, and multiple-angle formulas , $\sin(A \pm B)$, $\cos 2A$, etc.
6. Product-to-sum and sum-to-product formulas , turning products into sums.
7. Trigonometric equations and general solutions , solving $\sin x = a$, $\cos x = a$, $\tan x = a$.

Key results / Formula card

| Conversion | $\pi$ rad $= 180^\circ$, so $1$ rad $= \frac{180^\circ}{\pi}$, $1^\circ = \frac{\pi}{180}$ rad | | Arc length | $s = r\theta$ ($\theta$ in radians) | | Pythagorean | $\sin^2\theta + \cos^2\theta = 1$ | | | $1 + \tan^2\theta = \sec^2\theta$ | | | $1 + \cot^2\theta = \csc^2\theta$ | | Sum / difference | $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ | | | $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ | | | $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ | | Double angle | $\sin 2A = 2 \sin A \cos A$ | | | $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ | | | $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$ | | Triple angle | $\sin 3A = 3 \sin A - 4 \sin^3 A$, $\cos 3A = 4\cos^3 A - 3\cos A$ | | Sum-to-product | $\sin C + \sin D = 2 \sin\!\frac{C+D}{2} \cos\!\frac{C-D}{2}$ | | | $\cos C + \cos D = 2 \cos\!\frac{C+D}{2} \cos\!\frac{C-D}{2}$ | | General solution | $\sin x = \sin \alpha \iff x = n\pi + (-1)^n \alpha$, $n \in \mathbb{Z}$ | | | $\cos x = \cos \alpha \iff x = 2n\pi \pm \alpha$ | | | $\tan x = \tan \alpha \iff x = n\pi + \alpha$ |

How to read this chapter

Sketch every graph by hand at least twice. Recite the sum-and-difference identities until they come back without thought. Practise solving 20+ trigonometric equations of the form $a \sin x + b \cos x = c$ , every JEE paper has one.