Trigonometry
Trigonometry tests the ratios sine, cosine, and tangent, standard angles, and heights-and-distances problems. SEE questions usually give an angle of elevation or depression and one length, then ask for a height, distance, or angle.
Core rules
- In a right triangle: sinθ=opposite/hypotenuse; cosθ=adjacent/hypotenuse; tanθ=opposite/adjacent.
- Standard values: sin30=1/2, cos30=√3/2, tan30=1/√3; sin45=cos45=1/√2, tan45=1; sin60=√3/2, cos60=1/2, tan60=√3.
- Complementary: sin(90°−θ)=cosθ, tan(90°−θ)=cotθ.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
- Draw a right triangle, label the known angle and side, then choose the ratio that connects known and unknown.
Worked example
A ladder 10 m long leans against a wall and makes 60° with the ground. The wall height is the side opposite the 60° angle, so sin60=height/10. Height=10×sin60=10×(√3/2)=5√3≈8.66 m. The distance of the foot from the wall would use cos60: 10×1/2=5 m. Choosing tan here would be wrong because the hypotenuse, not the adjacent side, is known.
Traps
- Measuring the angle of depression from the vertical instead of the horizontal.
- Using tan when the hypotenuse is involved.
- Forgetting to convert a calculator from degree to radian mode, or vice versa.
- Mixing exact surd answers with premature decimals.
- Not drawing the diagram, so opposite and adjacent are swapped.