Find the Exact Value sin(105)

Problem

sin(105)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

sin(105)=sin(60+45)

2. Apply the formula for the sine of a sum, 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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