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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