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 $1^{\circ }={\displaystyle \frac{\pi }{180}}$ 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 $(a,b)$: $\cos x=a$, $\sin x=b$, $\tan x={\displaystyle \frac{\sin x}{\cos x}}$, and $\cot ,\sec ,\mathrm{cosec}$ are the reciprocals. Hence ${\sin }^{2}x+{\cos }^{2}x=1$, $1+{\tan }^{2}x={\sec }^{2}x$, $1+{\cot }^{2}x={\mathrm{cosec}}^{2}x$.

• Signs: all positive in quadrant I, only $\sin$ (and $\mathrm{cosec}$) 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 $\mathbb{R}-\{(2n+1){\displaystyle \frac{\pi }{2}}\}$, range $\mathbb{R}$. $\sec x,\mathrm{cosec}x$: range $(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$.

4. Compound angles

$\sin (x\pm y)=\sin x\cos y\pm \cos x\sin y$; $\cos (x\pm y)=\cos x\cos y\mp \sin x\sin y$; $\tan (x\pm y)={\displaystyle \frac{\tan x\pm \tan y}{1\mp \tan x\tan y}}$. Allied angles follow: $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$, $\sin (\pi -x)=\sin x$, $\cos (\pi +x)=-\cos x$, and so on.

5. Multiple angles and sum-to-product

• $\sin 2x=2\sin x\cos x={\displaystyle \frac{2\tan x}{1+{\tan }^{2}x}}$; $\cos 2x={\cos }^{2}x-{\sin }^{2}x=2{\cos }^{2}x-1=1-2{\sin }^{2}x$; $\tan 2x={\displaystyle \frac{2\tan x}{1-{\tan }^{2}x}}$.
• $\sin 3x=3\sin x-4{\sin }^{3}x$, $\cos 3x=4{\cos }^{3}x-3\cos x$.
• $\sin A+\sin B=2\sin {\displaystyle \frac{A+B}{2}}\cos {\displaystyle \frac{A-B}{2}}$, $\sin A-\sin B=2\cos {\displaystyle \frac{A+B}{2}}\sin {\displaystyle \frac{A-B}{2}}$, $\cos A+\cos B=2\cos {\displaystyle \frac{A+B}{2}}\cos {\displaystyle \frac{A-B}{2}}$, $\cos A-\cos B=-2\sin {\displaystyle \frac{A+B}{2}}\sin {\displaystyle \frac{A-B}{2}}$.

Worked example 1

Convert ${225}^{\circ }$ to radians.

$225\times {\displaystyle \frac{\pi }{180}}={\displaystyle \frac{5\pi }{4}}$.

Worked example 2

Find the exact value of $\sin {\displaystyle \frac{\pi }{12}}$.

$\sin ({\displaystyle \frac{\pi }{3}}-{\displaystyle \frac{\pi }{4}})={\displaystyle \frac{\sqrt{3}}{2}}\cdot {\displaystyle \frac{\sqrt{2}}{2}}-{\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{\sqrt{2}}{2}}={\displaystyle \frac{\sqrt{6}-\sqrt{2}}{4}}$.

Worked example 3

If $\sin x={\displaystyle \frac{3}{5}}$ and $x$ is in quadrant II, find $\cos x$ and $\tan x$.

$\cos x=-{\displaystyle \frac{4}{5}}$ (negative in II), $\tan x=-{\displaystyle \frac{3}{4}}$.

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.