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Find the Exact Value sin(12)

Problem

sin(12)

Solution

  1. Identify the expression as the sine of an angle measured in degrees, sin(12)

  2. Apply the half-angle formula or angle subtraction formula. To find the exact value, we can use the identity sin(12)=sin(30−18)

  3. Use the sine subtraction formula sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

  4. Substitute the known values for 30 and 18 We know sin(30)=1/2 and cos(30)=√(,3)/2

  5. Recall the exact values for 18 sin(18)=(√(,5)−1)/4 and cos(18)=√(,10+2√(,5))/4

  6. Calculate the expression:

sin(12)=sin(30)*cos(18)−cos(30)*sin(18)

  1. Plug in the values:

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

  1. Simplify the fractions by finding a common denominator of 8

sin(12)=(√(,10+2√(,5))−√(,3)*(√(,5)−1))/8

  1. Distribute the √(,3) in the second term:

sin(12)=(√(,10+2√(,5))−√(,15)+√(,3))/8

Final Answer

sin(12)=(√(,10+2√(,5))+√(,3)−√(,15))/8

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