Trigonometry and Its Applications
This section covers Trigonometry and Its Applications. The exam tests trigonometric ratios, values for 0°, 30°, 45°, 60° and 90°, simple identities, and heights-and-distances problems.
Core rules
In a right triangle with angle θ:
- sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- tan θ = sin θ/cos θ and cot θ = 1/tan θ. Key values include sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, cos 30° = √3/2, and tan 45° = 1. In applications, the angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal.
Worked example
A ladder 10 m long makes an angle of 60° with the ground. How high does it reach? The height is the side opposite the 60° angle, so height = 10 sin 60° = 10×√3/2 = 5√3 m, approximately 8.66 m.
Traps
Do not use the sloping length as the opposite side. Angles of elevation and depression are measured from the horizontal, not from the vertical. Check that the calculator or table value matches the quadrant and angle. In identities, work on one side until it equals the other rather than assuming equality.