Trigonometry in right-angled triangles
The three ratios (SOH-CAH-TOA)
- Name the sides from the angle $\theta$: opposite (across), adjacent (next to), hypotenuse.
$$\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}$$
- To find an angle, use the inverse: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.
- Worked example: opp $4$, adj $3$ → $\tan\theta = \tfrac43$, $\theta = \tan^{-1}(\tfrac43) = 53.1^{\circ}$.
Practice
In a right-angled triangle the hypotenuse is 10 cm and the angle is 30°. The opposite side = 10 × sin 30°. Find it (cm).
10 × sin 30° = 10 × 0.5 = 5 cm.
Practice
The opposite side is 4 cm and the adjacent side is 3 cm. Find the angle (degrees, 1 dp).
tan θ = 4/3, so θ = tan⁻¹(4/3) = 53.1°.
Elevation and depression (Extended)
- Angle of elevation: looking up from the horizontal. Angle of depression: looking down.
- Worked example: $50\ \text{m}$ from a tower, elevation $40^{\circ}$ → height $= 50\tan 40^{\circ} = 42.0\ \text{m}$.
Practice
From 50 m away, the angle of elevation to a tower top is 40°. Height = 50 × tan 40°. Find it (m, 1 dp).
50 × tan 40° = 50 × 0.839 = 42.0 m.
You've got it
Key idea
- SOH-CAH-TOA: $\sin = \tfrac{\text{opp}}{\text{hyp}}$, $\cos = \tfrac{\text{adj}}{\text{hyp}}$, $\tan = \tfrac{\text{opp}}{\text{adj}}$
- use the inverse ($\tan^{-1}$ etc.) to find an angle
- elevation looks up, depression looks down — both from the horizontal