CIE AMath Summary 2

Chapter 9: TRIGONOMETRIC FUNCTIONS

Chapter 9: TRIGONOMETRIC FUNCTIONS

Introduction

Trigonometric functions describe periodic phenomena such as waves, oscillations, and circular motion. This summary covers periodic behaviour, the sine, cosine, and tangent functions, trigonometric equations, identities, and solving equations in quadratic form.

Periodic Behaviour

Definition

A function $f(x)$ is periodic if there exists a positive constant $P$ (the period) such that:

$ f(x+P)=f(x) \quad \text{for all } x \text{ in the domain} $

The smallest such $P$ is the fundamental period.

Properties of Periodic Functions

• The graph repeats every $P$ units.
• If $f$ has period $P$, then $f(kx)$ has period $\frac{P}{|k|}$.
• The sum of periodic functions may not be periodic unless periods are commensurable.

Periods of Trigonometric Functions

Function	Principal Period
$\sin x$, $\cos x$, $\csc x$, $\sec x$	$2\pi$
$\tan x$, $\cot x$	$\pi$

Examples

1. $\sin(2x)$ has period $ \frac{2\pi}{2}=\pi $
2. $ \cos\left(\frac{x}{3}\right) $ has period $ \frac{2\pi}{1/3}=6\pi $
3. $ \tan(4x) $ has period $ \frac{\pi}{4} $
4. The function $ f(x)=\sin x+\sin(2x) $ is periodic with period $2\pi$ (least common multiple of $2\pi$ and $\pi$).

The Sine Function

Definition

$ y=\sin x $ is defined as the $y$-coordinate of the point on the unit circle at angle $x$ radians.

Key Properties

• Domain: $\mathbb{R}$ (all real numbers)
• Range: $[-1,1]$
• Period: $2\pi$
• Amplitude: $1$
• Odd function: $\sin(-x)=-\sin x$
• Zeros: $x=n\pi$ where $n\in\mathbb{Z}$
• Maximum: $1$ at $x=\frac{\pi}{2}+2n\pi$
• Minimum: $-1$ at $x=\frac{3\pi}{2}+2n\pi$

Graph of $y=\sin x$

Click to view the Graph

Transformed Sine Function

For $ y=a\sin(bx-c)+d $ :

• Amplitude: $|a|$
• Period: $\frac{2\pi}{|b|}$
• Phase shift: $\frac{c}{b}$ (to the right if $c > 0$)
• Vertical shift: $d$

Examples

1. Find the amplitude, period, and phase shift of $ y=3\sin(2x-\pi) $.
   Amplitude $=3$, Period $ \frac{2\pi}{2}=\pi $, Phase shift $ \frac{\pi}{2} $ to the right.
2. Sketch $ y=\sin\left(x-\frac{\pi}{2}\right) $.
   This is $\sin x$ shifted right by $ \frac{\pi}{2} $ (equivalent to $-\cos x$).
3. Evaluate $ \sin\frac{5\pi}{6} $.
   $ \frac{5\pi}{6}=150^\circ $, $ \sin\frac{5\pi}{6} = \sin\frac{\pi}{6} = \frac{1}{2} $

The Cosine Function

Definition

$y=\cos x$ is defined as the $x$-coordinate of the point on the unit circle at angle $x$ radians.

Key Properties

• Domain: $\mathbb{R}$
• Range: $[-1,1]$
• Period: $2\pi$
• Amplitude: $1$
• Even function: $\cos(-x)=\cos x$
• Zeros: $x=\frac{\pi}{2}+n\pi$
• Maximum: $1$ at $x=2n\pi$
• Minimum: $-1$ at $x=\pi+2n\pi$

Graph of $y=\cos x$

Click to view the Graph

Relationship Between Sine and Cosine

$$ \sin x=\cos\left(x-\frac{\pi}{2}\right), \qquad \cos x=\sin\left(x+\frac{\pi}{2}\right) $$

Examples

1. Find the zeros of $y=2\cos(3x)$ on $[0,2\pi)$.

   $\cos(3x)=0 \implies 3x=\frac{\pi}{2}+n\pi \implies x=\frac{\pi}{6}+\frac{n\pi}{3}$

   For $n=0,1,2,3,4,5$: $$ x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} $$

2. Evaluate $\cos\frac{4\pi}{3}$.

   $\frac{4\pi}{3}=240^\circ$, reference angle $\frac{\pi}{3}$ (Quadrant III, cosine negative).

   $$ \cos\frac{4\pi}{3} = -\cos\frac{\pi}{3} = -\frac12 $$

3. Find the maximum and minimum of $y=5-3\cos x$.

   $$ \cos x\in[-1,1] $$ $$ 5-3\cos x \in [5-3(1),\,5-3(-1)] = [2,8] $$

   Maximum $=8$, Minimum $=2$.

The Tangent Function

Definition

$$ \tan x=\frac{\sin x}{\cos x} $$

Key Properties

• Domain: $x\neq \frac{\pi}{2}+n\pi$ (where $\cos x=0$)
• Range: $\mathbb{R}$
• Period: $\pi$
• Odd function: $\tan(-x)=-\tan x$
• Zeros: $x=n\pi$
• Vertical asymptotes: $x=\frac{\pi}{2}+n\pi$
• Increasing on each interval $\left(-\frac{\pi}{2}+n\pi,\frac{\pi}{2}+n\pi\right)$

Graph of $y=\tan x$

Click to view the Graph

Transformed Tangent Function

For $y=a\tan(bx-c)+d$:

• Period: $\frac{\pi}{|b|}$
• Phase shift: $\frac{c}{b}$
• Vertical shift: $d$
• Asymptotes at $bx-c=\frac{\pi}{2}+n\pi$

Examples

1. Find the period and asymptotes of $y=\tan(2x)$.

   Period: $$ \frac{\pi}{2} $$

   Asymptotes: $$ 2x=\frac{\pi}{2}+n\pi $$ $$ x=\frac{\pi}{4}+\frac{n\pi}{2} $$

2. Evaluate $\tan\frac{3\pi}{4}$.

   $\frac{3\pi}{4}=135^\circ$ (Quadrant II, tangent negative).

   Reference angle: $$ \frac{\pi}{4} $$ $$ \tan\frac{3\pi}{4} = -\tan\frac{\pi}{4} = -1 $$

3. Solve $\tan x=\sqrt3$ for $x\in[0,2\pi)$.

   Reference angle: $$ \frac{\pi}{3} $$

   Tangent is positive in Quadrants I and III:

   $$ x=\frac{\pi}{3}, \qquad x=\frac{4\pi}{3} $$

Trigonometric Equations

General Solution Formulas

$$ \sin\theta=\sin\alpha \implies \theta=\alpha+2n\pi \quad\text{or}\quad \theta=\pi-\alpha+2n\pi $$

$$ \cos\theta=\cos\alpha \implies \theta=\pm\alpha+2n\pi $$

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

Solving Strategy

1. Isolate the trigonometric function.
2. Find the reference angle (principal value).
3. Determine all quadrants where the function has the given sign.
4. Write the general solution using periodicity.
5. Find required solutions in the given interval.

Examples

1. Solve $\sin x=\frac{\sqrt3}{2}$ for $x\in[0,2\pi)$.

   Reference angle: $$ \frac{\pi}{3} $$

   Sine is positive in Quadrants I and II:

   $$ x=\frac{\pi}{3}, \qquad x=\frac{2\pi}{3} $$

2. Solve $\cos x=-\frac12$ for $x\in\mathbb{R}$.

   $$ x=\frac{2\pi}{3}+2n\pi $$ or $$ x=\frac{4\pi}{3}+2n\pi $$

3. Solve $\tan x=-1$ for $x\in[0,2\pi)$.

   $$ x=\frac{3\pi}{4}, \qquad x=\frac{7\pi}{4} $$

4. Solve $2\sin x+1=0$ for $x\in[0,2\pi)$.

   $$ \sin x=-\frac12 $$

   $$ x=\frac{7\pi}{6}, \qquad x=\frac{11\pi}{6} $$

Trigonometric Relationships (Identities)

Fundamental Identities

$$ \sin^2 x + \cos^2 x = 1 $$ $$ 1 + \tan^2 x = \sec^2 x $$ $$ 1 + \cot^2 x = \csc^2 x $$

Reciprocal Identities

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

Quotient Identities

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

Even-Odd Identities

$$ \sin(-x) = -\sin x \quad \text{(odd)} $$ $$ \cos(-x) = \cos x \quad \text{(even)} $$ $$ \tan(-x) = -\tan x \quad \text{(odd)} $$

Co-function Identities

$$ \sin\left(\frac{\pi}{2}-x\right)=\cos x,\quad \cos\left(\frac{\pi}{2}-x\right)=\sin x $$ $$ \tan\left(\frac{\pi}{2}-x\right)=\cot x $$

Addition and Subtraction Formulas

$$ \sin(A+B)=\sin A\cos B+\cos A\sin B $$ $$ \sin(A-B)=\sin A\cos B-\cos A\sin B $$ $$ \cos(A+B)=\cos A\cos B-\sin A\sin B $$ $$ \cos(A-B)=\cos A\cos B+\sin A\sin B $$ $$ \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} $$ $$ \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B} $$

Double Angle Formulas

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

Examples

1. Simplify $ \frac{\sin^2 x}{1-\cos x} $.

   $ \frac{\sin^2 x}{1-\cos x} = \frac{1-\cos^2 x}{1-\cos x} = \frac{(1-\cos x)(1+\cos x)}{1-\cos x} = 1+\cos x $

2. Prove $ \tan x \sin x + \cos x = \sec x $.

   LHS $ = \frac{\sin x}{\cos x}\cdot\sin x+\cos x = \frac{\sin^2 x}{\cos x}+\cos x = \frac{\sin^2 x+\cos^2 x}{\cos x} = \frac{1}{\cos x} = \sec x $

3. Find $ \cos 75^\circ $ exactly.

   $ \cos 75^\circ = \cos(45^\circ+30^\circ) = \cos45^\circ\cos30^\circ - \sin45^\circ\sin30^\circ $

   $ = \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} - \frac{\sqrt2}{2}\cdot\frac12 = \frac{\sqrt6-\sqrt2}{4} $

4. Simplify $ \sin(2x)\cos x+\cos(2x)\sin x $.

   Using the sine addition formula:

   $ \sin(2x)\cos x+\cos(2x)\sin x = \sin(2x+x) = \sin(3x) $

Trigonometric Equations in Quadratic Form

Method

Equations of the form $a\sin^2 x+b\sin x+c=0$ (or with $\cos x$ or $\tan x$) are solved by:

1. Substitute $u=\sin x$ (or $\cos x$, $\tan x$).
2. Solve the quadratic $au^2+bu+c=0$.
3. Back-substitute and solve for $x$.
4. Check for extraneous solutions (especially when squaring or using $\tan$).

Examples

1. Solve $2\sin^2 x-\sin x-1=0$ for $x\in[0,2\pi)$.

   Let $u=\sin x$:

   $ 2u^2-u-1=0 $

   $ (2u+1)(u-1)=0 \implies u=-\frac12 \text{ or } u=1 $

   $ \sin x=-\frac12 \Rightarrow x=\frac{7\pi}{6},\frac{11\pi}{6} $

   $ \sin x=1 \Rightarrow x=\frac{\pi}{2} $

   Solutions: $ x=\frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} $

2. Solve $2\cos^2 x-3\cos x+1=0$ for $x\in[0,2\pi)$.

   Let $u=\cos x$:

   $ 2u^2-3u+1=0 $

   $ (2u-1)(u-1)=0 \implies u=\frac12 \text{ or } u=1 $

   $ \cos x=\frac12 \Rightarrow x=\frac{\pi}{3},\frac{5\pi}{3} $

   $ \cos x=1 \Rightarrow x=0 $

   Solutions: $ x=0, \frac{\pi}{3}, \frac{5\pi}{3} $

3. Solve $\tan^2 x-3=0$ for $x\in[0,2\pi)$.

   $ \tan^2 x=3 \Rightarrow \tan x=\pm\sqrt3 $

   $ \tan x=\sqrt3 \Rightarrow x=\frac{\pi}{3}, \frac{4\pi}{3} $

   $ \tan x=-\sqrt3 \Rightarrow x=\frac{2\pi}{3}, \frac{5\pi}{3} $

   Solutions: $ x= \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} $

4. Solve $\sin^2 x-\cos^2 x=0$ for $x\in[0,2\pi)$.

   Using $ \cos^2 x=1-\sin^2 x $

   $ \sin^2 x-(1-\sin^2 x)=0 $

   $ 2\sin^2 x-1=0 \Rightarrow \sin^2 x=\frac12 \Rightarrow \sin x=\pm\frac{\sqrt2}{2} $

   Solutions: $ x= \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $

5. Solve $3\sec^2 x-4=0$ for $x\in[0,2\pi)$.

   $ \sec^2 x=\frac43 \Rightarrow \cos^2 x=\frac34 \Rightarrow \cos x=\pm\frac{\sqrt3}{2} $

   $ \cos x=\frac{\sqrt3}{2} \Rightarrow x=\frac{\pi}{6}, \frac{11\pi}{6} $

   $ \cos x=-\frac{\sqrt3}{2} \Rightarrow x=\frac{5\pi}{6}, \frac{7\pi}{6} $

   Solutions: $ x= \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} $

Using Identities to Create Quadratic Form

Sometimes the equation needs to be rewritten using identities before substitution.

Example: Using $ \cos 2x $ Identity

Solve $ \cos 2x+\cos x=0 $ for $x\in[0,2\pi)$.

Using $ \cos 2x=2\cos^2 x-1 $:

$$ 2\cos^2 x-1+\cos x=0 \implies 2\cos^2 x+\cos x-1=0 $$

Let $u=\cos x$:

$$ 2u^2+u-1=0 \implies (2u-1)(u+1)=0 $$ $$ u=\frac12 \quad\text{or}\quad u=-1 $$

$ \cos x=\frac12 \Rightarrow x=\frac{\pi}{3}, \frac{5\pi}{3} $

$ \cos x=-1 \Rightarrow x=\pi $

Solutions: $ x= \frac{\pi}{3}, \pi, \frac{5\pi}{3} $

Summary Table

Concept	Key Points
Periodicity	$\sin$, $\cos$ period $2\pi$; $\tan$ period $\pi$
Sine	Domain $\mathbb{R}$, range $[-1,1]$, odd
Cosine	Domain $\mathbb{R}$, range $[-1,1]$, even
Tangent	Domain $x \neq \frac{\pi}{2}+n\pi$, range $\mathbb{R}$, odd
General solutions	$\sin\theta=\sin\alpha \Rightarrow \theta=\alpha+2n\pi$ or $\pi-\alpha+2n\pi$
	$\cos\theta=\cos\alpha \Rightarrow \theta=\pm\alpha+2n\pi$
	$\tan\theta=\tan\alpha \Rightarrow \theta=\alpha+n\pi$
Pythagorean identities	$\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$
Double angle	$\sin2\theta=2\sin\theta\cos\theta$, $\cos2\theta=\cos^2\theta-\sin^2\theta$
Quadratic equations	Substitute $u=\sin x$ (or $\cos x$, $\tan x$), solve, back-substitute