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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