Find the Exact Value cos(24)

Problem

cos(24)

Solution

1. Identify the goal as finding the exact value of cos(24) using known trigonometric constants.

2. Apply the sum formula for cosine, cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) by expressing 24 as 60−36

3. Recall the exact values for 60 cos(60)=1/2 and sin(60)=√(,3)/2

4. Recall the exact values for 36 cos(36)=(1+√(,5))/4 and sin(36)=√(,10−2√(,5))/4

5. Substitute these values into the identity cos(60−36)=cos(60)*cos(36)+sin(60)*sin(36)

cos(24)=(1/2)*((1+√(,5))/4)+(√(,3)/2)*(√(,10−2√(,5))/4)

6. Simplify the expression by combining the terms over a common denominator of 8

cos(24)=(1+√(,5)+√(,30−6√(,5)))/8

Final Answer

cos(24)=(1+√(,5)+√(,30−6√(,5)))/8

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