Find the Exact Value sin(105 degrees )

Problem

sin(105^∘)

Solution

1. Rewrite the angle 105^∘ as a sum of two special angles from the unit circle, such as 60^∘ and 45^∘

sin(105^∘)=sin(60^∘+45^∘)

2. Apply the sum formula for sine, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

sin(60^∘+45^∘)=sin(60^∘)*cos(45^∘)+cos(60^∘)*sin(45^∘)

3. Substitute the exact values for the sine and cosine of 60^∘ and 45^∘

sin(60^∘)=√(,3)/2

cos(45^∘)=√(,2)/2

cos(60^∘)=1/2

sin(45^∘)=√(,2)/2

4. Multiply the terms together.

sin(105^∘)=√(,3)/2⋅√(,2)/2+1/2⋅√(,2)/2

sin(105^∘)=√(,6)/4+√(,2)/4

5. Combine the fractions over a common denominator.

sin(105^∘)=(√(,6)+√(,2))/4

Final Answer

sin(105^∘)=(√(,6)+√(,2))/4

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