Introduction to Trigonometry and Its Applications
This domain tests the six trigonometric ratios, standard angles, identities, and heights-and-distances problems using angles of elevation and depression.
Core rules
- In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; cosec, sec and cot are reciprocals.
- Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2, cos 60° = 1/2.
- Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
- In heights-and-distances problems, draw the right triangle, mark the known side and angle, then choose the ratio that connects them.
Worked example
A ladder 10 m long leans against a vertical wall, making 60° with the ground. How high up the wall does it reach? The wall height is opposite the 60° angle and the ladder is the hypotenuse. sin 60° = h/10, so h = 10 × √3/2 = 5√3 m ≈ 8.66 m. The distance of the foot from the wall is 10 cos 60° = 5 m, consistent with the 30°–60°–90° triangle.
Traps
- Using the angle of depression inside the triangle without converting it to the matching angle of elevation.
- Mixing up opposite and adjacent sides.
- Forgetting that sin²θ means (sin θ)², not sin(θ²).
- Leaving a surd answer unrationalised when the paper expects a simplified form.