Trigonometry: Ratios, Standard Angles and Heights
What the exam tests
Trigonometry questions test right-triangle ratios, the standard angles 30°, 45° and 60°, the identity sin²θ+cos²θ=1, and applications to heights and distances using angle of elevation or depression.
Core rules
- sin θ=opposite/hypotenuse; cos θ=adjacent/hypotenuse; tan θ=opposite/adjacent.
- tan θ=sin θ/cos θ; sin²θ+cos²θ=1.
- Standard values: sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2; cos values reverse.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
- In a right triangle, choose the ratio that connects the known side and the required side.
Worked example
A ladder 10 m long makes a 60° angle with the ground. The height reached is the side opposite the 60° angle, so height =10 sin 60°=10×√3/2=5√3≈8.66 m. The distance of the foot from the wall is 10 cos 60°=5 m. These two results satisfy 5²+(5√3)²=100.
Traps
Mixing opposite and adjacent is the main error; mark the angle first. Do not use the ladder length as the vertical height. Keep the calculator in degree mode. The identity is sin²θ+cos²θ=1, not sin θ+cos θ=1. Angles of elevation and depression are measured from the horizontal, not from the vertical wall. Keep units consistent and simplify surds exactly where possible.