Trigonometry Ratios — Sin, Cos, and Tan Explained
Trigonometric ratios — sine (sin), cosine (cos), and tangent (tan) — relate the angles of a right triangle to the ratios of its sides. Once you know an angle and one side, you can find all other sides. Once you know two sides, you can find all angles.
The mnemonic SOH-CAH-TOA summarizes the three ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. This guide explains each ratio, the unit circle, special angle values, and practical applications.
Formula
sin(θ) = O/H cos(θ) = A/H tan(θ) = O/A
SOH-CAH-TOA — The Three Basic Ratios
SOH: Sin(θ) = Opposite / Hypotenuse. The sine of an angle is the ratio of the side opposite the angle to the hypotenuse.
CAH: Cos(θ) = Adjacent / Hypotenuse. The cosine of an angle is the ratio of the adjacent side (the one that forms the angle but is not the hypotenuse) to the hypotenuse.
TOA: Tan(θ) = Opposite / Adjacent. The tangent is the ratio of the opposite side to the adjacent side. Note: tan(θ) = sin(θ)/cos(θ).
Important: these ratios only apply to right triangles, and the angle θ must be one of the two acute angles (not the 90° angle). The hypotenuse is always opposite the right angle — it never changes. "Adjacent" and "Opposite" switch roles depending on which acute angle you label as θ.
Special Angle Values — The Trig Table You Must Know
These values appear constantly in geometry and physics problems. Memorize or derive them from the 30-60-90 and 45-45-90 triangles.
| Angle (degrees) | Angle (radians) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 = 0.500 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | π/4 | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | π/3 | √3/2 ≈ 0.866 | 1/2 = 0.500 | √3 ≈ 1.732 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 ≈ 0.866 | −1/2 = −0.500 | −√3 ≈ −1.732 |
| 135° | 3π/4 | √2/2 ≈ 0.707 | −√2/2 ≈ −0.707 | −1 |
| 150° | 5π/6 | 1/2 = 0.500 | −√3/2 ≈ −0.866 | −1/√3 ≈ −0.577 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Deriving Special Angles Without Memorization
45-45-90 triangle: Start with a square with side 1. Cut it diagonally. Each right triangle has legs 1 and 1, hypotenuse √2. So sin(45°) = cos(45°) = 1/√2 = √2/2 ≈ 0.707, and tan(45°) = 1.
30-60-90 triangle: Start with an equilateral triangle with side 2. Cut it down the middle height. Each right triangle has hypotenuse 2, short leg 1 (half the base), and long leg √3 (by Pythagoras: √(4−1) = √3). So: sin(30°) = 1/2, cos(30°) = √3/2; sin(60°) = √3/2, cos(60°) = 1/2.
The pattern: Notice that sin(θ) and cos(90°−θ) are always equal — they're "co-functions." sin(30°) = cos(60°) = 1/2. sin(45°) = cos(45°) = √2/2. This symmetry means you only need to memorize half the table.
Inverse Trig Functions — Finding Angles
To find an angle from a ratio, use the inverse functions: arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹). These are available on every scientific calculator.
arcsin: If sin(θ) = x, then θ = arcsin(x). Range: −90° to 90°.
arccos: If cos(θ) = x, then θ = arccos(x). Range: 0° to 180°.
arctan: If tan(θ) = x, then θ = arctan(x). Range: −90° to 90°.
Example: a right triangle with opposite = 4, hypotenuse = 5. sin(θ) = 4/5 = 0.8. θ = arcsin(0.8) ≈ 53.13°. You can verify: cos(53.13°) ≈ 0.6 = 3/5, and adjacent = 5 × 0.6 = 3. Check: 3² + 4² = 9+16 = 25 = 5² ✓
Practical Applications of Trig Ratios
Finding a side: If you know angle θ and the hypotenuse H, the opposite side = H × sin(θ) and the adjacent side = H × cos(θ). Example: a 30-ft ladder leaning at 70° to the ground. Height reached = 30 × sin(70°) ≈ 28.2 ft.
Finding height indirectly: Stand 100 m from a building. Look up at 35° to the top. Height ≈ 100 × tan(35°) ≈ 70.0 m. This is how surveyors measure building heights.
Navigation and bearings: A ship sails 50 km at a bearing of 40° north of east. Northward distance = 50 × cos(40°) ≈ 38.3 km. Eastward distance = 50 × sin(40°) ≈ 32.1 km.