Find the Exact Value csc((5pi)/12)

Problem

csc((5*π)/12)

Solution

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

(5*π)/12=(2*π)/12+(3*π)/12

2. Simplify the fractions to standard angles.

(5*π)/12=π/6+π/4

3. Apply the reciprocal identity for cosecant.

csc((5*π)/12)=1/sin(π/6+π/4)

4. Use the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

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

5. Substitute the exact values for the trigonometric functions.

sin(π/6+π/4)=(1/2)*(√(,2)/2)+(√(,3)/2)*(√(,2)/2)

6. Simplify the expression for sine.

sin((5*π)/12)=√(,2)/4+√(,6)/4

sin((5*π)/12)=(√(,2)+√(,6))/4

7. Calculate the reciprocal to find the cosecant value.

csc((5*π)/12)=4/(√(,6)+√(,2))

8. Rationalize the denominator by multiplying the numerator and denominator by the conjugate √(,6)−√(,2)

csc((5*π)/12)=(4*(√(,6)−√(,2)))/((√(,6)+√(,2))*(√(,6)−√(,2)))

csc((5*π)/12)=(4*(√(,6)−√(,2)))/(6−2)

csc((5*π)/12)=(4*(√(,6)−√(,2)))/4

csc((5*π)/12)=√(,6)−√(,2)

Final Answer

csc((5*π)/12)=√(,6)−√(,2)

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