Expand Using Sum/Difference Formulas cos((5pi)/12)

Problem

cos((5*π)/12)

Solution

1. Identify two angles from the unit circle that sum or subtract to (5*π)/12 We can use (3*π)/12+(2*π)/12 which simplifies to π/4+π/6

2. Apply the formula for the cosine of a sum, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

3. Substitute the values A=π/4 and B=π/6 into the formula.

cos(π/4+π/6)=cos(π/4)*cos(π/6)−sin(π/4)*sin(π/6)

4. Evaluate the trigonometric functions using known values from the unit circle.

cos(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

sin(π/4)=√(,2)/2

sin(π/6)=1/2

5. Multiply the fractions.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Subtract the results to find the final value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

cos((5*π)/12)=(√(,6)−√(,2))/4

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