Trigonometry: Ratios, Rules and Identities
What you must command
Trigonometry questions mix right-angled triangles with general triangles. For right-angled work, know sine, cosine and tangent as ratios of sides, and choose the ratio by which sides are involved. Be able to find a missing side or angle, and handle angles of elevation and depression. For non-right triangles, know the sine rule, the cosine rule and the area formula using half the product of two sides times the sine of the included angle. Know the exact values for 30, 45 and 60 degrees, the graphs and ranges of the trigonometric functions, and the identity linking sine squared and cosine squared. Work in degrees unless radians are stated.
Traps that catch candidates
- Choose the ratio from the sides you have and the side you want, not from habit.
- When finding an angle, use the inverse function, and check that your calculator is in degree mode.
- The sine rule has an ambiguous case for angles; the cosine rule avoids it when finding an angle from three sides.
- The area formula needs the angle between the two sides used, not any angle.
- Angles of depression are measured downward from the horizontal, so they often equal an angle of elevation inside the triangle.
- Sine and cosine values lie between -1 and 1; an answer outside that signals an error.
Worked example
A triangle has sides 7 cm and 9 cm with an included angle of 40 degrees. Area = half times 7 times 9 times sine 40 degrees. Sine 40 is about 0.643, so area is about 20.2 square centimetres.
Rule of thumb
Label the triangle first, marking which side is opposite which angle. Then match labels to the formula; most trigonometry errors are labelling errors, not calculation errors.