Simplify sin(165)+sin(105)

Problem

sin(165)+sin(105)

Solution

1. Identify the sum-to-product formula for sine, which states sin(A)+sin(B)=2*sin((A+B)/2)*cos((A−B)/2)

2. Substitute the values A=165 and B=105 into the formula.

sin(165)+sin(105)=2*sin((165+105)/2)*cos((165−105)/2)

3. Simplify the arguments inside the sine and cosine functions.

(165+105)/2=270/2=135

(165−105)/2=60/2=30

4. Rewrite the expression using the simplified angles.

sin(165)+sin(105)=2*sin(135)*cos(30)

5. Evaluate the trigonometric values using the unit circle.

sin(135)=√(,2)/2

cos(30)=√(,3)/2

6. Multiply the resulting values together.

2⋅√(,2)/2⋅√(,3)/2=√(,6)/2

Final Answer

sin(165)+sin(105)=√(,6)/2

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