Every right triangle with the same acute angle has the same shape, so the ratios of its sides do not depend on its size. Trigonometry gives those ratios names and lets you use them to find lengths and angles you cannot measure directly. In Geometry, this grows out of similar triangles and ends with real problems about heights, ramps and distances.
Where the ratios come from
Draw a right triangle with an acute angle of 30°. Any other right triangle with a 30° angle is similar to it, so corresponding sides are in the same proportion. That is why the ratio of two sides is fixed by the angle alone. We name the three most useful ratios, using the sides relative to angle A:
- sin A = opposite ÷ hypotenuse
- cos A = adjacent ÷ hypotenuse
- tan A = opposite ÷ adjacent
The hypotenuse is the side across from the right angle. The opposite side is across from angle A, and the adjacent side is the leg that touches A. A common memory aid is SOH-CAH-TOA.
A first example with exact values
Take a right triangle with legs 3 and 4 and hypotenuse 5 (check: 9 + 16 = 25). For the angle A opposite the side of length 3:
- sin A = 3 ÷ 5 = 0.6
- cos A = 4 ÷ 5 = 0.8
- tan A = 3 ÷ 4 = 0.75
To find the angle itself, use the inverse tangent on a calculator: A = tan⁻¹(0.75) ≈ 36.9°. The other acute angle is 90° − 36.9° = 53.1°, and for that angle the ratios swap: its sine is 0.8 and its cosine is 0.6.
Sine and cosine of complementary angles
That swap is no accident. The sine of an angle equals the cosine of its complement: sin A = cos(90° − A). For example, sin 20° ≈ 0.3420 and cos 70° ≈ 0.3420. It happens because the side opposite one acute angle is the side adjacent to the other.
Finding a missing side
Worked example 1: tree height. You stand 30 feet from the base of a tree and measure an angle of elevation of 35° to the top. The side you know (30 ft) is adjacent to the angle, and the side you want (the height h) is opposite. The ratio that links opposite and adjacent is tangent:
tan 35° = h ÷ 30, so h = 30 × tan 35° ≈ 30 × 0.7002 ≈ 21.0 feet.
Worked example 2: a ladder. A 12-foot ladder leans against a wall at 70° to the ground. The ladder is the hypotenuse.
- Height reached on the wall (opposite 70°): 12 × sin 70° ≈ 12 × 0.9397 ≈ 11.3 feet.
- Distance of the foot from the wall (adjacent to 70°): 12 × cos 70° ≈ 12 × 0.3420 ≈ 4.1 feet.
Check with the Pythagorean theorem: 11.28² + 4.10² ≈ 127.2 + 16.8 = 144, and 12² = 144.
Worked example 3: unknown hypotenuse. The angle is 28° and the adjacent side is 15. Cosine relates adjacent and hypotenuse, so cos 28° = 15 ÷ x. Then x = 15 ÷ cos 28° ≈ 15 ÷ 0.8829 ≈ 17.0. Because the unknown is in the denominator, you divide.
Finding a missing angle
Worked example 4: a ramp. A ramp rises 1.5 meters over a horizontal run of 12 meters. The rise is opposite the angle and the run is adjacent, so tan θ = 1.5 ÷ 12 = 0.125. Then θ = tan⁻¹(0.125) ≈ 7.1°.
A routine for any problem
- Draw the triangle and mark the right angle.
- Mark the angle you know or want, then label hypotenuse, opposite and adjacent relative to that angle.
- Decide which two sides are involved (one known, one unknown, or two known).
- Pick the ratio that uses those two sides and write the equation.
- Solve, then check that the answer is sensible: the hypotenuse must be the longest side.
Common mistakes
- Calculator in the wrong mode. If the angle is in degrees, the calculator must be in degree mode. In radian mode, sin 30 gives about −0.988 instead of 0.5.
- Labeling sides from the wrong angle. Opposite and adjacent change when you switch to the other acute angle.
- Multiplying when you should divide. If the unknown is the denominator, divide.
- Using these ratios on triangles without a right angle. SOH-CAH-TOA is for right triangles only.
- Rounding too early. Keep extra decimals until the final step.
Practice Geometry questions with an explanation for every answer, or read the guide to California Geometry practice.
Source: the California Common Core State Standards for Mathematics, G-SRT.6, G-SRT.7, G-SRT.8.