Find the Exact Value sec(pi/12)

Problem

sec(π/12)

Solution

1. Identify the angle π/12 as a difference of two special angles from the unit circle, such as π/3−π/4 or π/4−π/6 We will use π/4−π/6

2. Apply the reciprocal identity for secant, which states that sec(θ)=1/cos(θ)

3. Substitute the difference of angles into the cosine function:

cos(π/4−π/6)

4. Apply the cosine difference formula cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

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

5. Evaluate the trigonometric values for the special angles:

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

6. Simplify the expression for the cosine:

cos(π/12)=(√(,6)+√(,2))/4

7. Calculate the reciprocal to find the secant value:

sec(π/12)=4/(√(,6)+√(,2))

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

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

9. Simplify the denominator using the difference of squares (a+b)*(a−b)=a2−b2

sec(π/12)=(4*(√(,6)−√(,2)))/(6−2)

10. Finalize the simplification by canceling the common factor of 4:

sec(π/12)=√(,6)−√(,2)

Final Answer

sec(π/12)=√(,6)−√(,2)

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