8.2.2 Law of Sines and Law of Cosines (continued)
The Law of Cosines is used in this example to find the measure of an angle in a triangle. Notice that the Law of Sines cannot be used in this example because there is no pair of known angle measures or known opposite side length to set up in a proportion. The Law of Cosines can be used because all three side lengths are known. If necessary, label the sides as a , b , and c to help set up the equation for the Law of Cosines correctly. Substitute the side lengths, and the unknown, m ∠ Q , into the Law of Cosines. Isolate cos ∠ Q and solve for the other side of the equation. m ∠ Q = cos − 1 (0.76) ≈ 41 ° . The length of the side of a triangle and the measure of one angle of the triangle are determined in this application example. The measures of two sides and one angle are given, but the length of the side opposite the known angle is unknown. Begin by using the Law of Cosines to find the length of the rope (the unknown side of the triangle). Substitute the length of the unknown side, a , and the known values into the Law of Cosines. Solve for a 2 , take the square root of both sides, and solve for the positive square root to find a . The length of the rope is approximately 2.9 meters. To find the measure of the angle the rope makes with the ground, use the Law of Sines, since there is a known angle with a known opposite side length. Set up the proportion sin 91 ° is to 2.9 m as sin B (the unknown angle) is to 2.25 m. Cross multiply and rearrange to isolate sin B . Calculate the value of sin B on a calculator. To find m ∠ B , find the inverse sine of the value for sin B . The result gives m ∠ B ≈ 51 ° .
Example 4 Engineering Example
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