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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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