Law of sines

The law of sines, also called sine rule or sine formula, lets you find missing measures in a triangle when you know the measures of two angles and a side, or two sides and a nonincluded angle.

Law of sines

Looking closely at the triangle above, did you make the following observations?

Side a is opposite to angle A. Side b is opposite to angle B. Side c is opposite to angle C.

Now, how do we know the formula will work?

Now, do this useful experiment below that will really help you understand the sine rule. You will need a piece of paper, a pencil, a protractor, and a scientific calculator.

1. Draw a scalene triangle on a sheet of paper and label the triangle as the one I did above.

2. Use a ruler to measure sides a, b, and c.

3. Use a protractor to measure angle A, angle B, and angle C.

4. Use the sine rule to verify that it works.

I did the same thing for the triangle I drew above and I have found the following measurements.

a = 13 cm , b = 13.7 cm, and c = 9.3 cm

A = 66 degrees, B = 75 degrees, and C = 42 degrees

sin(66°) / 13 cm
= 0.0702

sin(75°) / 13.7 cm
= 0.0705

sin(42°) / 9.3 cm
= 0.0701

As you can see, the answers are almost the same. If our measurements were perfect, they will be exactly the same.

How to use the law of sines to solve a problem

Use the triangle above and the law of sines to find the length of x and the length of y.

sin(63°) / 18 m
= 0.0495

0.0495 =
sin(71°) / x

Multiply both sides by x

x × 0.0495 = sin(71°)

Divide both sides by 0.0495

x =
sin(71°) / 0.0495
= 19.1 m

Before we can find y, we need to know the measure of the angle that is opposite to y. Call this angle n.

Since the sum of the angle in a triangle is 180°,  63 + 71 + n = 180

134 + n = 180, so  n = 46°

0.0495 =
sin(46°) / y

Multiply both sides by y

y × 0.0495 = sin(46°)

Divide both sides by 0.0495

y =
sin(46°) / 0.0495
= 14.53 m

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