# 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.

Notice that side a is opposite to angle A. Also, side b is opposite to angle B. Now, how do we know the formula will work?

Do this experiment

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

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°)

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°)

y =
sin(46°) / 0.0495
= 14.53 m

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