Trigonometry and Its Applications
The exam tests trigonometric ratios, values at 0°, 30°, 45°, 60° and 90°, standard identities, complementary angles, and heights-and-distances problems using elevation and depression.
Core rules
- In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; reciprocals are cosec, sec and cot.
- Values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
- Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ.
- Angles of elevation and depression are measured from the horizontal.
Worked example
A tower casts a shadow 30 m long when the angle of elevation of the sun is 30°. Let h be the height. Then tan 30° = h/30, so h = 30 × (1/√3) = 10√3 ≈ 17.3 m.
Traps
Using tan where cot is needed; measuring the angle of depression from the vertical; swapping the values of 30° and 60°; cancelling sin θ as if it were a factor; forgetting to rationalise 1/√3 or to convert units.