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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