Laws of sines & cosines
Solve any triangle, not just right ones.
Pen and paper is fine · no calculator needed why?
Opens at level 36. You're level 1. You can read and practice here now.
the idea
The right-triangle ratios stretch to any triangle. Name the angles A, B and C, and give each side the small letter of the angle across from it: side a faces angle A.
Two sides and the angle between them give the area, ½ab sin C, and the third side, by the law of cosines. The law of sines says each side divided by the sine of its opposite angle gives the same number.
techniques
Area with sine
- Area = ½ × a × b × sin C.
- sin 30° and sin 150° are both ½, and sin 90° is 1.
- With 90°, this is half of base times height.
worked example
A triangle has sides 6 and 9 with a 150° angle between them. What is its area?
- sin 150° = sin 30° = ½.
- Area = ½ × 6 × 9 × ½ = 13.5.
Answer: 13.5
Law of cosines
- c² = a² + b² − 2ab cos C.
- cos 60° = ½, so subtract ab. cos 90° = 0, so c² = a² + b².
- cos 120° = −½, so add ab.
- Take the square root last.
worked example
A triangle has sides 3 and 5 with a 120° angle between them. How long is the third side?
- cos 120° = −½, so add 3 × 5 = 15.
- c² = 9 + 25 + 15 = 49.
- c = √49 = 7.
Answer: 7
Law of sines
- Each side ÷ the sine of its opposite angle gives the same number.
- So b = a × sin B ÷ sin A: the side you want goes with its own angle.
- Look up the sines, multiply, then divide. Round at the end.
worked example
In triangle ABC, side a is opposite angle A and side b is opposite angle B. If a = 6, A = 30° and B = 45°, how long is side b? Round to 2 decimal places.
- b = 6 × sin 45° ÷ sin 30°.
- sin 45° ≈ 0.7071 and sin 30° = 0.5.
- 6 × 0.7071 ÷ 0.5 = 8.4852, which rounds to 8.49.
Answer: 8.49
watch out for
- Leaving out the ½ at the front of the area formula.
- Getting the sign of the cosine term wrong. At 120° the cosine is negative, so you add ab.
- Forgetting the square root. The law of cosines gives c², not c.
- Flipping the law of sines. The side you want goes with the sine of its own angle, on top.
practice
Area with sine
worked example
A triangle has sides 13 and 7 with a 90° angle between them. What is its area?
Answer: 45.5
- Area = ½ × a × b × sin C, and sin 90° = 1.
- Area = ½ × 13 × 7 × 1 = 45.5.
Law of cosines
worked example
A triangle has sides 10 and 16 with a 60° angle between them. How long is the third side?
Answer: 14
- c² = a² + b² − 2ab cos C, and cos 60° = ½.
- c² = 10² + 16² − 2 × 10 × 16 × ½ = 100 + 256 − 160 = 196.
- c = √196 = 14.
Law of sines
worked example
In triangle ABC, side a is opposite angle A and side b is opposite angle B. If a = 14, A = 45° and B = 90°, how long is side b? Round to 2 decimal places.
Answer: 19.8
- b ÷ sin B = a ÷ sin A, so b = 14 × sin 90° ÷ sin 45°.
- sin 90° = 1 and sin 45° ≈ 0.7071, so b ≈ 19.7990.
- Rounded to 2 decimal places: 19.80.