8.2.2 Law of Sines and Law of Cosines (continued)
The measure of an angle in a triangle is obtained in this example using the Law of Sines. The Law of Sines is used to set up the proportion sin G is to the length of the opposite known side, EF , as sin F is to the length of the opposite known side, EG . Substitute the known values, cross multiply, and rearrange to isolate sin F . Calculate the value of sin F on a calculator. To find m ∠ F , calculate the inverse sine of the value for sin F . The result gives m ∠ F ≈ 52 ° .
Example 3 Using the Law of Cosines
The Law of Cosines applies to any triangle, not just right triangles. For any △ ABC with side lengths a , b , and c , a 2 = b 2 + c 2 − 2 bc cos A , b 2 = a 2 + c 2 − 2 ac cos B , and c 2 = a 2 + b 2 − 2 ab cos C , where a is the length of the side opposite ∠ A , etc. The Law of Cosines is used in this example to find the length of the side of 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. If necessary, label the sides as a , b , and c to help set up the equation for the Law of Cosines correctly. Substitute y for the length of the unknown side, XZ , and substitute the known values into the Law of Cosines. Isolate y 2 , take the square root of both sides, and solve for the positive square root to find y .The length of side XZ is approximately 9.6 units.
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