Find the Exact Value sin(42)

Problem

sin(42^∘)

Solution

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

2. Recognize that 42^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘.

3. Apply the sum or difference formulas if possible. Note that 42^∘ can be expressed as 72^∘−30^∘ or 12^∘+30^∘ but these involve non-standard angles like 72^∘ or 12^∘

4. Use the exact value derived from the properties of a regular pentagon or the golden ratio ϕ=(1+√(,5))/2 where sin(18^∘)=(√(,5)−1)/4 and cos(36^∘)=(√(,5)+1)/4

5. Expand using the addition formula sin(42^∘)=sin(60^∘−18^∘)

6. Substitute the known values: sin(60^∘)=√(,3)/2 cos(60^∘)=1/2 sin(18^∘)=(√(,5)−1)/4 and cos(18^∘)=√(,10+2√(,5))/4

7. Calculate the result: sin(60^∘)*cos(18^∘)−cos(60^∘)*sin(18^∘)

8. Simplify the expression into a single fraction.

Final Answer

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

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