Find the Exact Value cos(42)

Problem

cos(42^∘)

Solution

1. Identify the angle as 42^∘ Since this is not a standard reference angle, we can express it using the sum or difference of standard angles or use the half-angle identity.

2. Apply the difference formula for cosine, cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) by choosing A=72^∘ and B=30^∘ or use the identity cos(42^∘)=sin(48^∘)

3. Use the exact value for sin(18^∘)=(√(,5)−1)/4 and cos(18^∘)=√(,10+2√(,5))/4 to derive values for 72^∘ and 48^∘

4. Substitute the known values into the expression. The exact value of cos(42^∘) is derived from the properties of a regular pentagon and trigonometric identities.

5. Simplify the resulting radical expression to its standard exact form.

Final Answer

cos(42^∘)=(√(,5)+1)/8√(,10+2√(,5))+√(,3)/8*(√(,5)−1)

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