Expand Using Sum/Difference Formulas sin(165)

Problem

sin(165)

Solution

1. Identify two angles from the unit circle that sum or subtract to 165 We can use 120 and 45

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

3. Substitute A=120 and B=45 into the formula.

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

4. Evaluate the trigonometric values for the specific angles.

sin(120)=√(,3)/2

cos(45)=√(,2)/2

cos(120)=−1/2

sin(45)=√(,2)/2

5. Multiply the terms together.

√(,3)/2⋅√(,2)/2=√(,6)/4

−1/2⋅√(,2)/2=−√(,2)/4

6. Combine the fractions to find the final value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(165)=(√(,6)−√(,2))/4

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