How to Find the Measure of the Two Missing Angles When One Is Given?

Use the law of sines to find the two missing angles of a triangle.

A "trick of the trade" is a shortcut or clever ruse that the knowledgeable practitioner uses to make his work or employment easier. Trigonometry is a trade, and geometricians have their own tricks of the trade. One of these is the law of sines. Like its counterpart, the law of cosines -- which allows you to calculate the length of the side of a triangle opposite a known angle and the other two sides, the law of sines allows you to calculate two missing angles when you know only one angle and two sides of the triangle, one of those known sides being opposite the known angle.

Things You'll Need

• Ruler
• Scientific calculator

Instructions

1. Place the triangle on the table in front of you so that the known angle is on your right and one side of the triangle is parallel to the edge of the table in front of you.

2. Label the known angle "B," label the next angle moving in a counterclockwise direction around the triangle "C" and label the remaining angle "A."

3. Label the side of the triangle closest to the front of the table "c," label the next side moving in a counterclockwise direction around the triangle "a" and label the remaining side of the triangle "b."

4. Measure the length of the sides you have labeled "a" and "b" using a ruler. For example, side a is 3 inches long and side b is 7 inches long.

5. Calculate the sine of the measurement of the known angle, which is angle B, using a scientific calculator. For example, if angle B measures 17 degrees, then sin (17) = 0.29237.

6. Multiply your answer by the quotient of the length of side a and side b using your calculator. For example, 0.29237 x (3/7) = 0.12530.

7. Take the arcsine of your answer using your calculator. For example, arcsin (0.12530) = 7.1982. The measurement of angle A is 7.1982 degrees.

8. Subtract the sum of angles A and B from 180. For example, 180 - (17 + 7.1982) = 155.80. The measurement of the remaining angle, angle C, is 155.80 degrees.

References

• Georgia State University; HyperPhysics; Law of Sines; C. R. Nave
• Georgia State University; HyperPhysics; Law of Cosines; C. R. Nave
• Temple College: 8.1 -- The Law of Sines
• "Calculus: Concepts and Contexts"; James Stewart; 2006