doing math

courses › Trigonometry

Laws of sines & cosines

level 36 course

Solve any triangle, not just right ones.

Pen and paper is fine · no calculator needed why?

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Opens at level 36.

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Builds on: Radians & the unit circle (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

You know two sides and the angle between them.

  1. Area = ½ × a × b × sin C.
  2. sin 30° and sin 150° are both ½, and sin 90° is 1.
  3. With 90°, this is half of base times height.
worked example

Example: A triangle has sides 6 and 9 with a 150° angle between them. What is its area?

  1. sin 150° = sin 30° = ½.
  2. Area = ½ × 6 × 9 × ½ = 13.5.

Answer: 13.5

Law of cosines

You know two sides and the angle between them, and want the third side.

  1. c² = a² + b² − 2ab cos C.
  2. cos 60° = ½, so subtract ab. cos 90° = 0, so c² = a² + b².
  3. cos 120° = −½, so add ab.
  4. Take the square root last.
worked example

Example: A triangle has sides 3 and 5 with a 120° angle between them. How long is the third side?

  1. cos 120° = −½, so add 3 × 5 = 15.
  2. c² = 9 + 25 + 15 = 49.
  3. c = √49 = 7.

Answer: 7

Law of sines

You know a side, the angle opposite it, and one more angle.

  1. Each side ÷ the sine of its opposite angle gives the same number.
  2. So b = a × sin B ÷ sin A: the side you want goes with its own angle.
  3. Look up the sines, multiply, then divide. Round at the end.
worked example

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.

  1. b = 6 × sin 45° ÷ sin 30°.
  2. sin 45° ≈ 0.7071 and sin 30° = 0.5.
  3. 6 × 0.7071 ÷ 0.5 = 8.4852, which rounds to 8.49.

Answer: 8.49

Tips by skill

  • TipArea with sine: Area is ½ × a × b × sin C. sin 30° and sin 150° are both ½; sin 90° is 1.
  • TipLaw of cosines: c² = a² + b² − 2ab cos C. Subtract ab at 60°, add ab at 120°, nothing at 90°. Then take the square root.
  • TipLaw of sines: b = a × sin B ÷ sin A: the side you want goes on top with its own angle. sin 120° = sin 60°; sin 135° = sin 45°.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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