4.4 Right-triangle trigonometry

Right-Triangle Trigonometry

A right triangle contains one 90° angle. The side opposite that angle — the longest side — is the hypotenuse. The other two sides are the legs. Two tools unlock nearly every right-triangle problem: the Pythagorean theorem for side lengths and the three trigonometric ratios for connecting sides and angles.

The Pythagorean theorem

For legs a and b and hypotenuse c: a² + b² = c². Use it to find a missing side when you know the other two.

Worked example 1: legs 6 and 8, find the hypotenuse.

c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10.

Worked example 2: hypotenuse 13, one leg 5, find the other leg.

5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = √144 = 12.

When you know the hypotenuse, subtract; when you know both legs, add.

The three trig ratios: SOH-CAH-TOA

For an acute angle, label the sides from that angle's point of view: the opposite side faces it, the adjacent side touches it (and is not the hypotenuse), and the hypotenuse lies across from the right angle.

Solving for a side with trig

Worked example 3: a right triangle has a 30° angle and a hypotenuse of 12. Find the side opposite the 30° angle.

Opposite and hypotenuse point to sine: sin 30° = opp / 12. Since sin 30° = 0.5, opp = 12 × 0.5 = 6.

Worked example 4: from a 40° angle, the adjacent leg is 10. Find the opposite leg.

Opposite and adjacent point to tangent: tan 40° = opp / 10. With tan 40° ≈ 0.839, opp = 10 × 0.839 = 8.39.

Solving for an angle

When you know two sides, an inverse trig function recovers the angle.

Worked example 5: the opposite side is 7 and the hypotenuse is 14. Then sin θ = 7/14 = 0.5, so θ = sin⁻¹(0.5) = 30°. Likewise, if the opposite is 5 and the adjacent is 5, then tan θ = 5/5 = 1, and θ = tan⁻¹(1) = 45°.

Special right triangles

Two right triangles appear so often that their side ratios are worth memorizing.

In a 45-45-90 triangle the two legs are equal and the hypotenuse is a leg times √2. Worked example 6: if each leg is 5, the hypotenuse is 5√2 ≈ 5 × 1.414 = 7.07.

In a 30-60-90 triangle, the side opposite 30° is the shortest; the side opposite 60° is that short side times √3; and the hypotenuse is twice the short side. Worked example 7: if the short leg (opposite 30°) is 4, then the long leg = 4√3 ≈ 6.93 and the hypotenuse = 2 × 4 = 8. These agree with the trig values: sin 30° = short/hyp = 4/8 = 0.5, exactly as expected.

Putting it together

Worked example 8: a 15 ft ladder leans against a wall with its base 9 ft from the wall. How high up the wall does it reach? The ladder is the hypotenuse.

9² + h² = 15² → 81 + h² = 225 → h² = 144 → h = 12 ft.

If instead you were told the ladder makes a 60° angle with the ground and is 15 ft long, the height is opposite that angle, so sin 60° = h/15, giving h = 15 × 0.866 = 12.99 ≈ 13 ft.

Using cosine and Pythagorean triples

Cosine handles the adjacent side. Worked example 9: a 50° angle has a hypotenuse of 20; find the adjacent leg. Adjacent and hypotenuse point to cosine: cos 50° = adj / 20, and with cos 50° ≈ 0.643, adj = 20 × 0.643 = 12.86.

A few whole-number right triangles, called Pythagorean triples, recur so often that recognizing them saves time: 3-4-5, 5-12-13, 8-15-17, and their multiples such as 6-8-10 (the very first example scaled by 2) and 9-12-15. If a problem gives two of these numbers as the legs, or as a leg and the hypotenuse, you can often read off the third side without a full calculation and then confirm it with a² + b² = c². This pattern recognition is faster than reaching for a calculator on every step.

Reasonableness and reminders

The hypotenuse is always the longest side, so any leg you compute must come out shorter than the hypotenuse — a leg longer than the hypotenuse means you added when you should have subtracted. Because sine and cosine are ratios of a leg to the hypotenuse, their values always fall between 0 and 1; a "sine" above 1 signals a setup error. Finally, decide opposite versus adjacent from the angle you are using: the same physical side is "opposite" one acute angle and "adjacent" to the other.