A-level Mathematics/CIE/Pure Mathematics 1/Trigonometry

The Trigonometric Functions

A right-angled triangle consists of three sides: the hypotenuse, the adjacent, and the opposite.

Sine

The sine of an angle is defined as the ratio between the opposite and the hypotenuse. For a given angle, this ratio will always be the same, even if the triangle is scaled up or down.

$\sin(\theta )={\frac {opposite}{hypotenuse}}$

Cosine

The cosine of an angle is defined as the ratio between the adjacent and the hypotenuse.

$\cos(\theta )={\frac {adjacent}{hypotenuse}}$

Tangent

The tangent of an angle is defined as the ratio between the opposite and the adjacent.

$\tan(\theta )={\frac {opposite}{adjacent}}$

The Unit Circle

The unit circle is a circle of radius 1. It can be used to provide an alternate way of looking at trigonometric functions.

In the unit circle, a right-angled triangle can be drawn with the radius as its hypotenuse. Thus, the hypotenuse is 1 and the sine and cosine functions refer to the coordinates of a point on the unit circle.

Sine and cosine correspond to the coordinates of a point on the unit circle

Graphing Trigonometric Functions

A sine wave

A sine graph starts at $(0,0)$, then oscillates with a period of $2\pi$ and an amplitude of $1$.

A cosine wave

A cosine graph is like a sine graph in that it oscillates with a period of $2\pi$ and an amplitude of $1$, but it starts at $(1,0)$

A tangent wave

A tangent graph starts at $(0,0)$, goes to infinity as it approaches $x={\frac {\pi }{2}}$, emerges from negative infinity after $x={\frac {\pi }{2}}$, then repeats this at $(\pi ,0)$. The tangent graph has a period of $\pi$.

Exact Values

It is useful to know the following exact values of trigonometric functions:

Exact Values

x/°	x/rad	sin x	cos x	tan x
0	0	0	1	0
30	π/6	1/2	√3/2	1/√3
45	π/4	1/√2	1/√2	1
60	π/3	√3/2	1/2	√3
90	π/2	1	0	undefined

Inverse Trigonometric Functions

A Note on Notation

Some sources may use $\arcsin$, $\arccos$, and $\arctan$ to represent the inverse trigonometric functions, but this notation is not endorsed by Cambridge International

The inverse trigonometric functions are functions that reverse the trigonometric functions, just like any other inverse function. The inverse trigonometric functions are: $\sin ^{-1}$, which is the inverse of $\sin$; $\cos ^{-1}$, which is the inverse of $\cos$; and $\tan ^{-1}$, which is the inverse of $\tan$.

Trigonometric Identities

An identity is a statement that is always true, such as $a+b\equiv b+a$. A trigonometric identity, therefore, is a trigonometric statement that is always true.

It is helpful to know the following identities:

• $\sin(x)\equiv \cos({\frac {\pi }{2}}-x)$
• $\cos(x)\equiv \sin({\frac {\pi }{2}}-x)$
• ${\frac {\sin(x)}{\cos {x}}}\equiv \tan(x)$
• $\sin ^{2}(x)+\cos ^{2}(x)\equiv 1$

These identities can be used to prove other identities.

e.g. Prove that ${\frac {\sin(x)\cos ^{2}(x)}{\cos(x)}}\equiv \tan(x)-\sin ^{2}(x)\tan(x)$

${\begin{aligned}{\frac {\sin(x)\cos ^{2}(x)}{\cos(x)}}&\equiv {\frac {\sin(x)(1-\sin ^{2}(x))}{\cos(x)}}\\&\equiv \tan(x)(1-\sin ^{2}(x))\\&\equiv \tan(x)-\sin ^{2}(x)\tan(x)\end{aligned}}$

Solving Trigonometric Equations

When solving a trigonometric equation, it is important to keep the interval in mind.

e.g. Solve ${\sqrt {3}}\tan 2x-1=0$ for $-{\frac {\pi }{2}}<x<{\frac {\pi }{2}}$.

${\begin{aligned}{\sqrt {3}}\tan 2x-1&=0\\{\sqrt {3}}\tan 2x&=1\\\tan 2x&={\frac {1}{\sqrt {3}}}\\&{\text{The interval is }}-{\frac {\pi }{2}}<x<{\frac {\pi }{2}}{\text{ so}}-\pi <2x<\pi \\2x&=\{-{\frac {5\pi }{6}},{\frac {\pi }{6}}\}\leftarrow {\text{We need to include all possible values that are in the interval}}\\x&=\{-{\frac {5\pi }{12}},{\frac {\pi }{12}}\}\end{aligned}}$