In a right triangle, for an acute angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. Their reciprocals are cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. Also tan θ = sin θ/cos θ.
Standard Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | not defined |
Remembering the table
sin for 0°, 30°, 45°, 60°, 90° is √0/2, √1/2, √2/2, √3/2, √4/2. cos is the same list backwards.
Identities
| Identity | Rearranged |
|---|---|
| sin²θ + cos²θ = 1 | sin²θ = 1 − cos²θ |
| 1 + tan²θ = sec²θ | sec²θ − tan²θ = 1, so (sec θ − tan θ)(sec θ + tan θ) = 1 |
| 1 + cot²θ = cosec²θ | cosec²θ − cot²θ = 1 |
- Complementary angles: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ. So sin 35°/cos 55° = 1 and tan 20° × tan 70° = 1.
- If sin A = cos B for acute angles, then A + B = 90°.
- From one ratio, draw the triangle: tan θ = 3/4 means opposite 3, adjacent 4, hypotenuse 5, so sin θ = 3/5 and cos θ = 4/5.
- For acute angles, sin θ and tan θ increase with θ, and cos θ decreases. sin θ and cos θ never exceed 1.
Degrees and radians
180° = π radians
Radians = degrees × π/180. 60° = π/3 and 1 radian ≈ 57.3°.
Heights and Distances
- The angle of elevation is measured upward from the horizontal to an object above; the angle of depression is measured downward to an object below. The two are equal for the same line of sight (alternate angles).
- Height = distance × tan(angle of elevation). A tower seen at 60° from 20 m away is 20√3 m tall.
- At 45° the height equals the distance. The shadow of a pole equals its height when the sun's elevation is 45°.
Common Mistakes
Swapping sin and cos values for 30° and 60°; writing sin²θ as sin θ²; and measuring the angle of depression from the vertical instead of the horizontal.