Find the Exact Value cos(48)

Problem

cos(48)

Solution

1. Identify the expression as the cosine of 48 degrees. Since 48 is not a standard angle, we look for a way to express it using standard angles or half-angle/double-angle identities.

2. Recognize that 48 can be related to 72 and 120 or more simply, it is part of the sequence of angles related to the regular pentagon (18,36,54,72.

3. Apply the sum identity cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) using 48=30+18

4. Substitute the known values for cos(30)=√(,3)/2 and sin(30)=1/2

5. Substitute the exact values for 18 where sin(18)=(√(,5)−1)/4 and cos(18)=√(,10+2√(,5))/4

6. Simplify the resulting expression:

cos(48)=cos(30)*cos(18)−sin(30)*sin(18)

cos(48)=√(,3)/2⋅√(,10+2√(,5))/4−1/2⋅(√(,5)−1)/4

cos(48)=(√(,30+6√(,5))−√(,5)+1)/8

Final Answer

cos(48)=(√(,30+6√(,5))−√(,5)+1)/8

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