Find the Exact Value cos(12)

Problem

cos(12^∘)

Solution

1. Identify the expression as the cosine of an angle measured in degrees.

2. Apply the half-angle formula or sum/difference identities. To find the exact value of cos(12^∘) we can use the identity cos(12^∘)=cos(72^∘−60^∘)

3. Use the difference identity for cosine, which is cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

4. Substitute the known values for 72^∘ and 60^∘ Note that cos(72^∘)=sin(18^∘)=(√(,5)−1)/4 and sin(72^∘)=cos(18^∘)=√(,10+2√(,5))/4

5. Evaluate the expression:

cos(12^∘)=cos(72^∘)*cos(60^∘)+sin(72^∘)*sin(60^∘)

6. Plug in the exact trigonometric constants:

cos(12^∘)=((√(,5)−1)/4)*(1/2)+(√(,10+2√(,5))/4)*(√(,3)/2)

7. Simplify the fractions by multiplying the numerators and denominators:

cos(12^∘)=(√(,5)−1)/8+√(,30+6√(,5))/8

8. Combine the terms over a common denominator:

cos(12^∘)=(√(,5)−1+√(,30+6√(,5)))/8

Final Answer

cos(12^∘)=(√(,5)−1+√(,30+6√(,5)))/8

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