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^∘)

7. Plug in the values:

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

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

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

9. 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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