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

Want more problems? Check here!