Expand Using Sum/Difference Formulas sin((3pi)/8)

Problem

sin((3*π)/8)

Solution

1. Identify the angle as a sum of two angles with known trigonometric values. We can write (3*π)/8 as π/8+(2*π)/8 but it is more useful to use π/4+π/8 or recognize that (3*π)/8=π/2−π/8 However, to use standard sum/difference formulas for expansion, we can use (3*π)/8=π/4+π/8

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

3. Substitute A=π/4 and B=π/8 into the formula.

sin(π/4+π/8)=sin(π/4)*cos(π/8)+cos(π/4)*sin(π/8)

4. Evaluate the known values for sin(π/4) and cos(π/4)

sin(π/4)=√(,2)/2

cos(π/4)=√(,2)/2

5. Combine the terms to show the expanded form.

sin((3*π)/8)=√(,2)/2*cos(π/8)+√(,2)/2*sin(π/8)

Final Answer

sin((3*π)/8)=√(,2)/2*(cos(π/8)+sin(π/8))

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