60-second revision
- sin 30° = 1/2, sin 45° = 1/sqrt(2), sin 60° = sqrt(3)/2; cos runs the same values in reverse.
- sin²A + cos²A = 1; sec²A − tan²A = 1; cosec²A − cot²A = 1.
- sec A + tan A = k means sec A − tan A = 1/k.
- sin A = cos B (both acute) means A + B = 90°.
- a sin A + b cos A lies between −sqrt(a² + b²) and +sqrt(a² + b²).
- tan 30° = 1/sqrt(3), tan 45° = 1, tan 60° = sqrt(3).
- Angle of depression from the top = angle of elevation from the bottom.
The standard-angle table
| Ratio | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/sqrt(2) | sqrt(3)/2 | 1 |
| cos | 1 | sqrt(3)/2 | 1/sqrt(2) | 1/2 | 0 |
| tan | 0 | 1/sqrt(3) | 1 | sqrt(3) | not defined |
Ratios from a right triangle
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent; cosec, sec and cot are their reciprocals. Given one ratio, draw the triangle and use Pythagoras.
Worked example: cos A = 5/13 means adjacent 5 and hypotenuse 13, so opposite = 12. Then sin A = 12/13 and tan A = 12/5.
The three identities
- sin²A + cos²A = 1
- sec²A − tan²A = 1, i.e. 1 + tan²A = sec²A
- cosec²A − cot²A = 1, i.e. 1 + cot²A = cosec²A
Worked example: (1 − sin²A) sec²A = cos²A × 1/cos²A = 1. Most simplification questions collapse to 1 with these identities.
Complementary angles
- sin (90° − A) = cos A and cos (90° − A) = sin A.
- tan (90° − A) = cot A and sec (90° − A) = cosec A.
- tan A × tan (90° − A) = 1.
- If sin x = cos y (both acute), then x + y = 90°.
Worked example: tan 20° × tan 40° × tan 50° × tan 70° = (tan 20° tan 70°)(tan 40° tan 50°) = 1 × 1 = 1. Worked example: if tan 2A = cot (A + 15°), then 3A + 15° = 90° and A = 25°.
Maximum and minimum
- a sin A + b cos A ranges from −sqrt(a² + b²) to +sqrt(a² + b²).
- a sin²A + b cos²A lies between the smaller and the larger of a and b.
- sin A and cos A each lie between −1 and 1.
Worked example: 8 sin A + 6 cos A has maximum sqrt(64 + 36) = 10 and minimum −10.
Heights and distances: the method
The angle of elevation is measured upwards from the horizontal; the angle of depression downwards. Because the horizontal lines are parallel, the angle of depression from the top equals the angle of elevation from the bottom.
- Draw a right triangle: height vertical, ground horizontal, line of sight as hypotenuse.
- Use tan when height and ground distance are involved; use sin or cos when a ladder, string or slant length is involved.
- Use the height:base ratios below to skip arithmetic.
| Angle | tan | height : base |
|---|---|---|
| 30° | 1/sqrt(3) | 1 : sqrt(3) |
| 45° | 1 | 1 : 1 |
| 60° | sqrt(3) | sqrt(3) : 1 |
Worked example: a kite string 80 m long makes 30° with the ground. Height = 80 × sin 30° = 40 m.
Worked example: the elevation of a tower top is 45° from a point and 60° after walking 20 m closer. With height h, h − h/sqrt(3) = 20, so h = 20 sqrt(3)/(sqrt(3) − 1) = 30 + 10 sqrt(3), about 47.3 m.
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