Expand Using Sum/Difference Formulas sin((7pi)/8)

Problem

sin((7*π)/8)

Solution

1. Identify a sum or difference of known angles that equals (7*π)/8 We can use the fact that (7*π)/8=(3*π)/4+π/8 or (7*π)/8=π−π/8 However, to use standard exact values, we can also view this as a half-angle or use (7*π)/8=π/2+(3*π)/8 To follow the instruction of sum/difference formulas specifically, we use (7*π)/8=(3*π)/4+π/8

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

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

sin((3*π)/4+π/8)=sin((3*π)/4)*cos(π/8)+cos((3*π)/4)*sin(π/8)

4. Evaluate the known trigonometric values for (3*π)/4

sin((3*π)/4)=√(,2)/2

cos((3*π)/4)=−√(,2)/2

5. Substitute these values back into the expanded expression.

sin((7*π)/8)=√(,2)/2*cos(π/8)−√(,2)/2*sin(π/8)

6. Factor out the common term.

sin((7*π)/8)=√(,2)/2*(cos(π/8)−sin(π/8))

Final Answer

sin((7*π)/8)=sin((3*π)/4)*cos(π/8)+cos((3*π)/4)*sin(π/8)

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