The law of sines

About this lesson

Integrated Math III › Trigonometric functions

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The law of sines

Warm-up

Three quick questions from earlier lessons.

Discover

In a triangle, the largest angle is opposite which side?

  1. The longest side
  2. The shortest side

Key idea

Sides are proportional to the sines of their opposite angles.

In any triangle, asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}. Use it when you know a side and its opposite angle plus one more part: two angles and a side, or two sides and an angle that is not between them (check for a second possible triangle).

Worked example

A = 30°, B = 45°, a = 6.

bsin⁡45°=6sin⁡30°\dfrac{b}{\sin 45°} = \dfrac{6}{\sin 30°}
Set up.
b=6⋅2/21/2=62b = 6 \cdot \dfrac{\sqrt2/2}{1/2} = 6\sqrt2
Solve.

Your turn, guided

Your turn: A = 30°, a = 5. What is a ÷ sin A?

1010
5 ÷ ½.

Practice ★

  1. You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
    1. When you know two angles and a side, such as A = 40°, B = 65° and a = 7
    2. When you know the three sides
    3. When you know two sides and the angle between them
  2. You know some parts of a triangle that has no right angle. When is the law of sines the right tool?
    1. When you know two angles and a side, such as A = 40°, B = 65° and a = 3
    2. When you know the three sides
    3. When you know two sides and the angle between them
  1. In triangle ABC, angle A = 30°, angle B = 45° and side a = 2 (opposite A). Use the law of sines to find side b (opposite B), exactly.

Watch out!

Practice ★★

  1. In triangle ABC, angle A = 30°, side a = 7 and side b = 11. Use the law of sines to find sin B.
  2. In triangle ABC, angle A = 30°, side a = 11 and side b = 10. Use the law of sines to find sin B.
  1. In triangle ABC, angle A = 30°, angle B = 45° and side a = 2 (opposite A). Use the law of sines to find side b (opposite B), exactly.

Exit ticket

Three questions to check you have got it.

  1. In triangle ABC, angle A = 30°, angle B = 45° and side a = 2 (opposite A). Use the law of sines to find side b (opposite B), exactly.
  2. In triangle ABC, angle A = 30°, angle B = 45° and side a = 6 (opposite A). Use the law of sines to find side b (opposite B), exactly.
  3. In triangle ABC, angle A = 30°, angle B = 45° and side a = 4 (opposite A). Use the law of sines to find side b (opposite B), exactly.

Summary card

Law of sines

a / sin A = b / sin B = c / sin C.

Watch out: pair each side with its opposite angle.

End of the lesson

Next lesson: The law of cosines Back to the chapter

Answers: The law of sines

A

  1. When you know two angles and a side, such as A = 40°, B = 65° and a = 7
  2. When you know two angles and a side, such as A = 40°, B = 65° and a = 3

B

  1. 222 \sqrt{2}

C

  1. 1114\frac{11}{14}
  2. 511\frac{5}{11}

D

  1. 222 \sqrt{2}

E

  1. 222 \sqrt{2}
  2. 626 \sqrt{2}
  3. 424 \sqrt{2}

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