7.1 Trigonometric functions

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).

22 2 The Greek letter θ\theta is often used for angles in the same way as the variable xx is used to denote unknowns.
cos(θ)=adjacenthypotenuse\displaystyle\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} (7.1)
sin(θ)=oppositehypotenuse\displaystyle\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} (7.2)
tan(θ)=oppositeadjacent\displaystyle\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} (7.3)

The way we work out the actual values of cos(θ)\cos(\theta), sin(θ)\sin(\theta) and tan(θ)\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


Note that because this is the unit circle, we have


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


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)\sin(x) and cos(x)\cos(x) as well as a diagram of the unit circle.