The six trigonometric ratios
In a right triangle, trigonometry links an angle to the ratios of the sides:
- sine = opposite ÷ hypotenuse
- cosine = adjacent ÷ hypotenuse
- tangent = opposite ÷ adjacent = sin ÷ cos
The other three are their reciprocals: cosecant = 1 ÷ sin, secant = 1 ÷ cos and cotangent = 1 ÷ tan. On the unit circle, these definitions extend to any angle, including negative angles and angles over 360°.
Degrees and radians
A full turn is 360° or 2π radians. To convert:
- radians = degrees × π ÷ 180
- degrees = radians × 180 ÷ π
So 30° = π/6 ≈ 0.5236 rad, 90° = π/2 and 180° = π. Calculators and spreadsheets often expect radians: forgetting to convert is the most common trigonometry mistake.
Worked example: 30°
- sin 30° = 0.5
- cos 30° = √3 ÷ 2 ≈ 0.866025
- tan 30° = 0.5 ÷ 0.866 ≈ 0.577350
- csc 30° = 2, sec 30° ≈ 1.154701, cot 30° ≈ 1.732051
- In radians, 30° is 0.523599.
Values of remarkable angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 ≈ 0.866 | √3/3 ≈ 0.577 |
| 45° | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | √3/2 ≈ 0.866 | 1/2 | √3 ≈ 1.732 |
| 90° | 1 | 0 | undefined |
| 180° | 0 | −1 | 0 |
Signs by quadrant
The calculator reduces the angle to between 0° and 360° and shows its quadrant. In quadrant I (0° to 90°) all ratios are positive; in II (90° to 180°) only the sine; in III (180° to 270°) only the tangent; in IV (270° to 360°) only the cosine. A handy mnemonic is “All Students Take Calculus”.
Useful identities
- sin² θ + cos² θ = 1
- tan θ = sin θ ÷ cos θ
- sin(90° − θ) = cos θ
- sin(2θ) = 2 sin θ cos θ
Practical uses
- Height of a tree or building: height = distance × tan(angle of elevation). At 20 m and 35°, the top is 20 × 0.700 = 14 m above eye level.
- Ramps and roofs: a slope angle θ climbs tan θ for every unit of horizontal run.
- Forces: a 100 N pull at 30° has a horizontal part of 100 × cos 30° = 86.6 N.