Find the Exact Value sin(15 degrees )

Problem

sin(15)

Solution

1. Rewrite the angle 15 as a difference of two common angles from the unit circle, such as 45 and 30

sin(15)=sin(45−30)

2. Apply the sine difference formula, which states sin(α−β)=sin(α)*cos(β)−cos(α)*sin(β)

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

3. Substitute the known exact values for the sine and cosine of 45 and 30

sin(45)=√(,2)/2

cos(30)=√(,3)/2

cos(45)=√(,2)/2

sin(30)=1/2

4. Multiply the fractions in each term.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

5. Subtract the resulting fractions to find the final exact value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(15)=(√(,6)−√(,2))/4

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