Question

Find the exact value of sec(-π/12). Show your work and keep the answer in the simplest form.

(A) √6 + √2
(B) 2√6 + 2√2
(C) √6 - √2
(D) 2√6 - 2√2

Answer 1

The exact value of sec(-π/12) is found by using the reciprocal property of secant and the cosine of a common angle, 15°. This results in an exact value of (√6 + √2) after simplification, which is an option (A).

Step-by-step explanation:

To find the exact value of sec(-π/12), we apply the properties of trigonometric functions. Given the fact that secant is the reciprocal of cosine, we can write sec(θ) = 1/cos(θ). Also, because secant is an even function, sec(-θ) = sec(θ). Thus, sec(-π/12) = sec(π/12). Next, we note that π/12 radians are equal to 15°, a common angle. To find the cosine of 15°, we can use the half-angle identity derived from the cos(2θ) identity: cos(15°) = cos(π/12) = √((1 + cos(30°))/2) = √((1 + √(3)/2)/2). After simplifying, we have: cos(π/12) = √((2 + √3)/4) = √(2/4) + √(3/4) = √2/2 + √3/2. Now, to find sec(π/12), take the reciprocal: sec(π/12) = 2/√2 + 2/√3 = √2 + √6. Hence, the exact value of sec(-π/12), by combining the terms, is (√6 + √2), which corresponds to option (A).