Find the Value Using the Unit Circle sin(105)

Problem

sin(105)

Solution

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

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

3. Substitute the values A=60 and B=45 into the formula.

sin(105)=sin(60+45)

sin(105)=sin(60)*cos(45)+cos(60)*sin(45)

4. Evaluate each trigonometric function using the unit circle values.

sin(60)=√(,3)/2

cos(45)=√(,2)/2

cos(60)=1/2

sin(45)=√(,2)/2

5. Multiply the terms together.

sin(105)=√(,3)/2⋅√(,2)/2+1/2⋅√(,2)/2

sin(105)=√(,6)/4+√(,2)/4

6. Combine the fractions over a common denominator.

sin(105)=(√(,6)+√(,2))/4

Final Answer

sin(105)=(√(,6)+√(,2))/4

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