Trigonometry: Ratios, Identities and Heights and Distances
Introduction to Trigonometry and Its Applications splits into ratio-and-identity work (1–3 marks) and heights-and-distances problems (3–5 marks).
Ratios, values and identities
For an acute angle θ in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, with cosec, sec and cot as the reciprocals. Learn the table for 0°, 30°, 45°, 60° and 90°; also tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. Complementary angles give sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.
Identities you must be able to derive: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Heights and distances
Draw the right triangle, mark the angle at the observer, and choose the ratio linking the known side to the unknown. The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal, so it equals the angle of elevation from the far point (alternate angles).
Worked example
From a point 30 m from the foot of a tower, the angle of elevation of the top is 30°. Find the height. tan 30° = h/30 → h = 30 × 1/√3 = 10√3 ≈ 17.3 m.
Traps that cost marks
- Using the adjacent side where the opposite is required - label the triangle before writing any ratio.
- tan 45° = 1 and tan 60° = √3 are swapped more often than any other pair.
- The angle of depression is taken from the horizontal, never from the vertical.
- In identity proofs, convert everything to sin and cos when stuck, and work on one side only.
- Take √3 ≈ 1.732 to three places and state the unit.