A calculus of the absurd

Chapter 7 Trigonometry

7.1 Trigonometric functions

You’ve probably come across the following formulae:⁴⁹⁴⁹ Often remembered using the sort-of mnemonic "SOH-CAH-TOA" (i.e. "COS=OPPOSITE/ADJACENT, COS=ADJACENT/HYPOTENUSE, TAN=OPPOSITE/ADJACENT).

⁵⁰⁵⁰ The Greek letter \(\theta \) is often used for angles in the same way as the variable \(x\) is used to denote unknowns.

\begin{align} & \cos (\theta ) = \frac {\text {adjacent}}{\text {hypotenuse}}\\ & \sin (\theta ) = \frac {\text {opposite}}{\text {hypotenuse}}\\ & \tan (\theta ) = \frac {\text {opposite}}{\text {adjacent}} \end{align}

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

\begin{align*} & \sin (\theta ) = \frac {y}{1} \\ & \cos (\theta ) = \frac {x}{1} \end{align*}

Note that because this is the unit circle, we have

\begin{equation*} x^2 + y^2 = 1 \end{equation*}

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

\begin{equation*} \cos ^2(\theta ) + \sin ^2(\theta ) = 1 \end{equation*}

Below you can find high-precision, to-scale plots of the graphs ⁵¹⁵¹ 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.