Expand Using Sum/Difference Formulas sin(105)

Problem

sin(105)

Solution

1. Identify two angles from the unit circle that sum to 105

105=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 known trigonometric values for 60 and 45

sin(60+45)=(√(,3)/2)*(√(,2)/2)+(1/2)*(√(,2)/2)

4. Multiply the fractions in each term.

sin(60+45)=√(,6)/4+√(,2)/4

5. Simplify by combining the terms over a common denominator.

sin(60+45)=(√(,6)+√(,2))/4

Final Answer

sin(105)=(√(,6)+√(,2))/4

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