Trigonometry and Heights and Distances
Two kinds of question appear: identities and standard-angle evaluation, and one or two height-and-distance problems solved by drawing a right triangle and choosing tan or cot. Deciding which vertex holds the right angle decides the mark.
Core rules
- sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, with sec, cosec and cot as reciprocals.
- Standard values: sin 0°, 30°, 45°, 60°, 90° = 0, ½, 1/√2, √3/2, 1; cos runs the same list backwards; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
- Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Angle of elevation is measured upward from the horizontal at the observer's eye; angle of depression is measured downward, and by alternate angles it equals the elevation of the observer as seen from the object.
Worked example
The elevation of a tower's top is 30° from a point on the ground. Moving 20 m towards the tower, the elevation becomes 60°. Find the height. Let x be the distance from the foot after moving: h = x tan 60° = √3x. Originally h = (x + 20) tan 30° = (x + 20)/√3. Equating: √3x = (x + 20)/√3, so 3x = x + 20, x = 10, and h = 10√3 ≈ 17.3 m.
Traps
- Using tan where cot is needed because the known side is the hypotenuse.
- Placing the right angle at the wrong point when the observer is on a building.
- Forgetting that a depression angle must be converted to an elevation inside the triangle.
- Leaving the answer as 10√3 when a decimal is demanded, or quoting √3 = 1.73 too coarsely.