Find the Exact Value sin(15)

Problem

sin(15^∘)

Solution

1. Identify the angle as a difference of two special angles from the unit circle, such as 45^∘ and 30^∘

2. Apply the formula for the sine of a difference, which is sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

3. Substitute A=45^∘ and B=30^∘ into the identity.

sin(45^∘−30^∘)=sin(45^∘)*cos(30^∘)−cos(45^∘)*sin(30^∘)

4. Evaluate the trigonometric functions using exact values: sin(45^∘)=√(,2)/2 cos(30^∘)=√(,3)/2 cos(45^∘)=√(,2)/2 and sin(30^∘)=1/2

sin(15^∘)=√(,2)/2⋅√(,3)/2−√(,2)/2⋅1/2

5. Simplify the expression by multiplying the fractions and combining the terms over a common denominator.

sin(15^∘)=√(,6)/4−√(,2)/4

sin(15^∘)=(√(,6)−√(,2))/4

Final Answer

sin(15^∘)=(√(,6)−√(,2))/4

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