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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