OddClasses — Odd ones figure it out.

Trigonometric Ratios

Refer to the diagram above, for a right-angled triangle with angle

\theta

\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}

💡 Memory trick: SOH CAH TOA

Theorem 1📌 Learn for exam

Unit Circle definition

Refer to the diagram below. On a unit circle (radius = 1), for any angle $\theta$ measured from the positive x-axis:

\cos\theta = x\text{-coordinate of }P \qquad \sin\theta = y\text{-coordinate of }P

(x, y) = (\cos \theta, \sin \theta)

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x} \qquad (x = \cos\theta \neq 0)

▸ Unit Circle — drag P around the circlethe point P is draggable

Loading diagram…

Special angles — memorise these:

\begin{array}{c|ccc} \theta & \sin\theta & \cos\theta & \tan\theta \\ \hline 0° & 0 & 1 & 0 \\ 30° & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{3}} \\ 45° & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 \\ 60° & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} \\ 90° & 1 & 0 & \text{undef} \end{array}

Tip

The unit circle definition works for all angles — not just angles in right-angled triangles. That is why we need it for Grade 12.

Worked Example

Without a calculator, find the exact value of $\sin 30° \cdot \cos 60° + \cos 30° \cdot \sin 60°$ .