Find the Exact Value sin(165)

Problem

sin(165)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

sin(165)=sin(120+45)

2. Apply the sum formula for sine, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

sin(120+45)=sin(120)*cos(45)+cos(120)*sin(45)

3. Substitute the known exact values for the trigonometric functions of 120 and 45

sin(120)=√(,3)/2

cos(45)=√(,2)/2

cos(120)=−1/2

sin(45)=√(,2)/2

4. Multiply the terms together.

sin(165)=(√(,3)/2)*(√(,2)/2)+(−1/2)*(√(,2)/2)

sin(165)=√(,6)/4−√(,2)/4

5. Simplify the expression by combining the fractions over a common denominator.

sin(165)=(√(,6)−√(,2))/4

Final Answer

sin(165)=(√(,6)−√(,2))/4

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