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

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

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

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 $-\frac{\pi }{2}<x<\frac{\pi }{2}$.

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