Find the Exact Value sin(105)-sin(15)

Problem

sin(105)−sin(15)

Solution

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

2. Substitute the values A=105 and B=15 into the formula.

sin(105)−sin(15)=2*cos((105+15)/2)*sin((105−15)/2)

3. Simplify the arguments inside the trigonometric functions.

(105+15)/2=120/2=60

(105−15)/2=90/2=45

4. Evaluate the trigonometric functions at the common angles 60 and 45

cos(60)=1/2

sin(45)=√(,2)/2

5. Multiply the resulting values together.

2⋅1/2⋅√(,2)/2=√(,2)/2

Final Answer

sin(105)−sin(15)=√(,2)/2

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