Trigonometry and Its Applications
Trigonometry and Its Applications tests the six ratios, standard-angle values, simple identities, and heights-and-distances problems. The exam expects exact surd values, not rounded decimals, unless stated.
Core rules
- In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; cosec, sec and cot are their reciprocals.
- Standard values include sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2, tan 30° = 1/√3.
- Identity: sin²θ + cos²θ = 1, with 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
Worked example
A tower casts a shadow 30 m long when the sun's elevation is 30°. Find the height h. tan 30° = h/30, so h = 30 × 1/√3 = 10√3 m ≈ 17.3 m. The tower is opposite the 30° angle and the shadow is adjacent, which is why tangent is the correct ratio.
Traps
Do not use sine when the known side is adjacent to the angle. The angle of depression from the top of a tower equals the angle of elevation from the ground point only when the lines are alternate angles to parallel horizontals. In identities, sin²θ means (sin θ)², not sin(θ²). Rationalise denominators such as 1/√3 when an exact answer is requested.