Factor by Grouping sin(15)

Problem

sin(15)

Solution

1. Identify the angle 15 as a difference of two common 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 values for the specific angles.

sin(45)=√(,2)/2

cos(30)=√(,3)/2

cos(45)=√(,2)/2

sin(30)=1/2

5. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Subtract the resulting fractions to find the final value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(15)=(√(,6)−√(,2))/4

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