Sides and Angles: Trig Ratio Calculations
Trigonometric ratios — sine, cosine, and tangent — relate the angles of a right triangle to the ratios of its sides. Remembered by the mnemonic SOH CAH TOA, these ratios allow you to calculate unknown sides or angles in right triangles, with applications in navigation, surveying, architecture, and physics.
Components of Trig Ratio Calculations
This section covers the three primary ratios and their inverses:
- Sine (sin): sin(θ) = opposite / hypotenuse — the ratio of the side opposite the angle to the longest side.
- Cosine (cos): cos(θ) = adjacent / hypotenuse — the ratio of the side next to the angle to the longest side.
- Tangent (tan): tan(θ) = opposite / adjacent — the ratio of the opposite side to the adjacent side.
- Inverse Trig Functions: Use sin⁻¹, cos⁻¹, or tan⁻¹ to find an angle when you know the ratio of two sides.
Examples of Trig Ratio Calculations
Sine Examples
- In a right triangle with angle 30°, the opposite side is 5 and the hypotenuse is 10. Check: sin(30°) = 5/10 = 0.5 ✓.
- Find the opposite side when the hypotenuse is 20 and the angle is 45°: opposite = 20 × sin(45°) = 20 × 0.7071 ≈ 14.14.
- A kite string is 50 meters long at an angle of 60° to the ground. The kite's height is 50 × sin(60°) = 50 × 0.866 ≈ 43.3 meters.
Cosine Examples
- In a right triangle with angle 60°, the adjacent side is 4 and the hypotenuse is 8. Check: cos(60°) = 4/8 = 0.5 ✓.
- Find the adjacent side when the hypotenuse is 15 and the angle is 30°: adjacent = 15 × cos(30°) = 15 × 0.866 ≈ 12.99.
- A ramp is 12 feet long at a 20° incline. The horizontal distance it covers is 12 × cos(20°) ≈ 12 × 0.9397 ≈ 11.28 feet.
Tangent Examples
- In a right triangle with angle 45°, the opposite and adjacent sides are both 7. Check: tan(45°) = 7/7 = 1 ✓.
- Find the opposite side when the adjacent side is 9 and the angle is 53°: opposite = 9 × tan(53°) ≈ 9 × 1.3270 ≈ 11.94.
- A tree casts a 15-meter shadow when the sun is at 40° elevation. The tree height is 15 × tan(40°) ≈ 15 × 0.8391 ≈ 12.59 meters.
Inverse Trig Examples
- A right triangle has opposite = 3 and hypotenuse = 5. The angle is sin⁻¹(3/5) = sin⁻¹(0.6) ≈ 36.87°.
- A right triangle has adjacent = 8 and hypotenuse = 10. The angle is cos⁻¹(8/10) = cos⁻¹(0.8) ≈ 36.87°.
- A ladder reaches 6 feet up a wall and its base is 4 feet from the wall. The angle is tan⁻¹(6/4) = tan⁻¹(1.5) ≈ 56.31°.