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

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.



Graphing Trigonometric Functions


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



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 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:

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

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

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

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