Find the Value Using the Unit Circle sin(165)

Problem

sin(165)

Solution

1. Identify the angle 165 as a sum or difference of special angles from the unit circle.

165=120+45

2. Apply the sine addition formula, which states sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

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

3. Determine the values of the trigonometric functions at these special angles using the unit circle.

sin(120)=√(,3)/2

cos(120)=−1/2

sin(45)=√(,2)/2

cos(45)=√(,2)/2

4. Substitute these values back into the addition formula.

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

5. Simplify the expression by multiplying the fractions and combining the terms.

sin(165)=√(,6)/4−√(,2)/4

sin(165)=(√(,6)−√(,2))/4

Final Answer

sin(165)=(√(,6)−√(,2))/4

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