7.1 Trigonometric functions ‣ Chapter 7 Trigonometry ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 7 Trigonometry ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

You’ve probably come across the following formulae:¹¹ 1 Often remembered using the sort-of mnemonic ”SOH-CAH-TOA” (i.e. ”COS=OPPOSITE/ADJACENT, COS=ADJACENT/HYPOTENUSE, TAN=OPPOSITE/ADJACENT).

	$\displaystyle\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$		(7.1)
	$\displaystyle\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$		(7.2)
	$\displaystyle\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$		(7.3)

The way we work out the actual values of $\cos(\theta)$ , $\sin(\theta)$ and $\tan(\theta)$ is by making things as easy as possible for ourselves; we draw a triangle inside a circle with radius one. From here, we know that

	$\displaystyle\sin(\theta)=\frac{y}{1}$
	$\displaystyle\cos(\theta)=\frac{x}{1}$

Note that because this is the unit circle, we have

x^{2}+y^{2}=1

And if we substitute $\cos(\theta)$ and $\sin(\theta)$ we get that

\cos^{2}(\theta)+\sin^{2}(\theta)=1

Below you can find high-precision, to-scale plots of the graphs ³³ 3 Protip: learn how to draw the graphs without having to thtink about it! of both $\sin(x)$ and $\cos(x)$ as well as a diagram of the unit circle.