Expand Using Sum/Difference Formulas sin(15)

Problem

sin(15^∘)

Solution

1. Rewrite the angle 15^∘ as a difference of two special angles from the unit circle whose sine and cosine values are known.

15^∘=45^∘−30^∘

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

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

3. Substitute the exact values for the trigonometric functions 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 terms to find the final exact value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

sin(15^∘)=(√(,6)−√(,2))/4

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