Trigonometric Functions Class 11: notes and important questions

Trigonometric Functions: revision notes

1. Angles in degrees and radians

One radian is the angle subtended at the centre by an arc equal to the radius. \(\pi \text{ rad} = 180^\circ\), so \(1^\circ = \dfrac{\pi}{180}\) rad. Arc length \(l = r\theta\) (\(\theta\) in radians). Positive angles are measured anticlockwise.

2. Trigonometric functions of any angle

If the ray at angle \(x\) meets the unit circle at \((a, b)\): \(\cos x = a\), \(\sin x = b\), \(\tan x = \dfrac{\sin x}{\cos x}\), and \(\cot, \sec, \operatorname{cosec}\) are the reciprocals. Hence \(\sin^2 x + \cos^2 x = 1\), \(1 + \tan^2 x = \sec^2 x\), \(1 + \cot^2 x = \operatorname{cosec}^2 x\).

• Signs: all positive in quadrant I, only \(\sin\) (and \(\operatorname{cosec}\)) in II, only \(\tan\) (and \(\cot\)) in III, only \(\cos\) (and \(\sec\)) in IV.
• Periods: \(\sin, \cos\): \(2\pi\); \(\tan\): \(\pi\). So \(\sin(2n\pi + x) = \sin x\).
• \(\sin(-x) = -\sin x\), \(\cos(-x) = \cos x\).

3. Domains, ranges and graphs

\(\sin x, \cos x\): domain \(\mathbb R\), range \([-1, 1]\). \(\tan x\): domain \(\mathbb R - \{(2n + 1)\dfrac\pi2\}\), range \(\mathbb R\). \(\sec x, \operatorname{cosec} x\): range \((-\infty, -1] \cup [1, \infty)\).

4. Compound angles

\(\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y\); \(\cos(x \pm y) = \cos x\cos y \mp \sin x\sin y\); \(\tan(x \pm y) = \dfrac{\tan x \pm \tan y}{1 \mp \tan x\tan y}\). Allied angles follow: \(\cos\left(\dfrac\pi2 - x\right) = \sin x\), \(\sin(\pi - x) = \sin x\), \(\cos(\pi + x) = -\cos x\), and so on.

5. Multiple angles and sum-to-product

• \(\sin 2x = 2\sin x\cos x = \dfrac{2\tan x}{1 + \tan^2 x}\); \(\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x\); \(\tan 2x = \dfrac{2\tan x}{1 - \tan^2 x}\).
• \(\sin 3x = 3\sin x - 4\sin^3 x\), \(\cos 3x = 4\cos^3 x - 3\cos x\).
• \(\sin A + \sin B = 2\sin\dfrac{A + B}2\cos\dfrac{A - B}2\), \(\sin A - \sin B = 2\cos\dfrac{A + B}2\sin\dfrac{A - B}2\), \(\cos A + \cos B = 2\cos\dfrac{A + B}2\cos\dfrac{A - B}2\), \(\cos A - \cos B = -2\sin\dfrac{A + B}2\sin\dfrac{A - B}2\).

Worked example 1

Convert \(225^\circ\) to radians.

\(225 \times \dfrac{\pi}{180} = \dfrac{5\pi}{4}\).

Worked example 2

Find the exact value of \(\sin\dfrac{\pi}{12}\).

\(\sin\left(\dfrac\pi3 - \dfrac\pi4\right) = \dfrac{\sqrt3}{2}\cdot\dfrac{\sqrt2}{2} - \dfrac12\cdot\dfrac{\sqrt2}{2} = \dfrac{\sqrt6 - \sqrt2}{4}\).

Worked example 3

If \(\sin x = \dfrac35\) and \(x\) is in quadrant II, find \(\cos x\) and \(\tan x\).

\(\cos x = -\dfrac45\) (negative in II), \(\tan x = -\dfrac34\).

Common errors

• Using degrees in \(l = r\theta\): \(\theta\) must be in radians.
• Sign of a ratio in quadrants II to IV taken from the acute-angle value only.
• \(\sin(x + y) = \sin x + \sin y\) (false), or \(\cos(x + y)\) with a plus sign between the products.
• \(\cos 2x = 2\cos x - 1\): the cosine is squared.

Exam tips

• Reduce a large angle by subtracting multiples of \(2\pi\) (or \(360^\circ\)) first, then use the quadrant sign.
• In identity proofs, start from the more complicated side and name the formula at each step.