Find the Exact Value sin(42 degrees )

Problem

sin(42^∘)

Solution

1. Identify the angle as 42^∘ Since this is not a standard reference angle, we can express it using the difference formula for sine: sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

2. Rewrite the angle using standard angles 72^∘ and 30^∘ or 60^∘ and 18^∘ Using 60^∘ and 18^∘ is effective because the exact values for 18^∘ are known: sin(18^∘)=(√(,5)−1)/4 and cos(18^∘)=√(,10+2√(,5))/4

3. Apply the formula for sin(60^∘−18^∘)

sin(60^∘−18^∘)=sin(60^∘)*cos(18^∘)−cos(60^∘)*sin(18^∘)

4. Substitute the known exact values:

sin(42^∘)=√(,3)/2⋅√(,10+2√(,5))/4−1/2⋅(√(,5)−1)/4

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

sin(42^∘)=(√(,30+6√(,5))−√(,5)+1)/8

Final Answer

sin(42^∘)=(√(,30+6√(,5))−√(,5)+1)/8

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