Find the Exact Value sin(75)-sin(15)

Problem

sin(75)−sin(15)

Solution

1. Identify the appropriate sum-to-product identity for the difference of two sines.

sin(A)−sin(B)=2*cos((A+B)/2)*sin((A−B)/2)

2. Substitute the given angles A=75 and B=15 into the identity.

sin(75)−sin(15)=2*cos((75+15)/2)*sin((75−15)/2)

3. Simplify the arguments inside the trigonometric functions.

(75+15)/2=45

(75−15)/2=30

4. Rewrite the expression using the simplified angles.

sin(75)−sin(15)=2*cos(45)*sin(30)

5. Evaluate the exact values of the trigonometric functions using the unit circle.

cos(45)=√(,2)/2

sin(30)=1/2

6. Multiply the values to find the final result.

2⋅√(,2)/2⋅1/2=√(,2)/2

Final Answer

sin(75)−sin(15)=√(,2)/2

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