SECTION 7.6 Double-angle and Half-angle Formulas 529 Finding Exact Values Using Half-angle Formulas Use a Half-angle Formula to find the exact value of: (a) ° cos15 (b) ( ) − ° sin 15 Solution EXAMPLE 6 (a) Because ° = ° 15 30 2 , use the Half-angle Formula for α cos 2 with α = ° 30 . Also, because ° 15 is in quadrant I, ° > cos15 0, so choose the + sign in using formula (11). ° = ° = + ° = + = + = + cos15 cos 30 2 1 cos30 2 1 3 2 2 2 3 4 2 3 2 (b) Use the fact that ( ) − ° = − ° sin 15 sin15 , and then use formula (10). ( ) − ° = − ° =− − ° =− − = − − = − − sin 15 sin 30 2 1 cos30 2 1 3 2 2 2 3 4 2 3 2 It is interesting to compare the answer found in Example 6(a) with the answer to Example 2 of Section 7.5. There it was calculated that π ( ) = ° = + cos 12 cos15 1 4 6 2 Based on this and the result of Example 6(a), ( ) + + 1 4 6 2 and 2 3 2 are equal. (Since each expression is positive, you can verify this equality by squaring each expression.) Two very different-looking, yet correct, answers can be obtained, depending on the approach taken to solve a problem. Now Work PROBLEM 21 Finding Exact Values Using Half-angle Formulas If α π α π = − < < cos 3 5 , 3 2 , find the exact value of: (a) α sin 2 (b) α cos 2 (c) α tan 2 Solution EXAMPLE 7 First, observe that if π α π < < 3 2 , then π α π < < 2 2 3 4 . As a result, α 2 lies in quadrant II. (a) Because α 2 lies in quadrant II, α > sin 2 0, so use the + sign in formula (10) to get α α ( ) = − = − − = = = = sin 2 1 cos 2 1 3 5 2 8 5 2 4 5 2 5 2 5 5 (b) Because α 2 lies in quadrant II, α < cos 2 0, so use the − sign in formula (11) to get α α ( ) = − + = − + − = − = − = − cos 2 1 cos 2 1 3 5 2 2 5 2 1 5 5 5 (continued)
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