Trigonometry: Ratios, Identities and Heights and Distances
This domain tests the six trigonometric ratios, the standard-angle table (0°, 30°, 45°, 60°, 90°), identities, complementary-angle relations and one- or two-triangle problems on heights and distances.
Ratios and identities
- sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent = sin θ/cos θ; cosec, sec and cot are the reciprocals.
- sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.
- Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3; sin 90° = 1, cos 90° = 0, tan 90° undefined.
Heights and distances
- The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal.
- Sketch one right triangle per object, mark the known distance or height, and choose the ratio containing the unknown.
Worked example
From a point 30 m from the foot of a vertical tower, the angle of elevation of the top is 30°. Find the height. Let h be the height. tan 30° = h/30, so h = 30 × (1/√3) = 10√3 ≈ 17.32 m. The horizontal distance is the adjacent side, so tan is the correct ratio, not sin.
Traps that cost marks
- Using sin when the adjacent side (the ground distance) is involved.
- Measuring the angle of depression from the vertical instead of the horizontal.
- Leaving 10√3 without a decimal when the question asks for an approximate height; use √3 ≈ 1.732.
- Writing tan 90° as a large number rather than 'undefined'.