The sine law can also be used to solve for unknown angles. The following example uses the sine law to help solve a surveying problem. Traditionally, surveying was based on a series of length and angle measurements. Once some values were known, calculations could be used to determine unknown values or to double check measurements.

A video demonstration of the solution for Example 2 is provided.

Sine Law Example

Example 2

A surveyor was working in a field between three points that form an obtuse triangle. She found the distances between the points, on either side of the obtuse angle, to be 161 m and 362 m. She found the acute angle adjacent to the 362 m distance to be 21°.

  1. What is the measure of the obtuse angle?

    Begin by drawing a diagram to represent the situation.

    The goal is to determine the value of B. Unfortunately, we don't know b so we cannot use the sine law directly to determine B. However, it is possible to determine A and then use the sum of the interior angles in a triangle to determine B.

    Now determine B using the sum of the interior angles in a triangle.

  2. What should the surveyor do if the angle she calculates does not match the measured angle?

    If the measured angle and the calculated angle don't match, there is an error somewhere. The surveyor will need to recheck her measurements and calculations and correct any errors until the two agree.

The sine law can be used multiple times to solve more complex problems. Pay attention to the multiple uses of the sine law in the next example.