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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