Expand the Trigonometric Expression sin(75)

Problem

sin(75)

Solution

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

sin(75)=sin(45+30)

2. Apply the formula for the sine of a sum, which is 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 known 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 terms together.

sin(75)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

sin(75)=√(,6)/4+√(,2)/4

5. Simplify the expression by combining the fractions over a common denominator.

sin(75)=(√(,6)+√(,2))/4

Final Answer

sin(75)=(√(,6)+√(,2))/4

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