# Law of sines

If any angle, its opposite side of a triangle and any third part of the triangle are known, then the other parts can be found using the law of sines. The law of sines says that in a triangle, the ratio of the length of the side to the sine of the opposite angle is constant.

The law of sines is obtained from the expression for the area of a triangle. If two sides are known and also the angle between them, the area of the triangle can be calculated. The area of the triangle above would be

We set the areas equal and then simplify the equation

The ratio of the sine of the angle to the length of the opposite side is constant in any given triangle. The inverse numbers are also equal. This is called the law of sines.

Example 1

Find x

We use the law of sines

Example 2

A plane sets off towards the northeast. After 2,8 kilometres, it turns directly east and continues its journey until it turns 161,6 degrees to return back to the airport. How far from the airport was the plane during the second turn?

There are 45 degrees between the northeast and the east, so the angle of the first turn becomes 135 degrees in the resulting triangle. The triangle angle of the second turn is 18,4 degrees.

The distance from the airport is 6,3 kilometres

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