Evaluate each expression. sec ( π 3 ) \sec\left(\frac{\pi}{3}\right) sec ( 3 π )

Here we're asked to evaluate the expression and the expression that we're given here is the secant of pi over three. Now remember the secant is the reciprocal of cosine. So here I can take one over my cosine value which I know is x on the unit circle. So I can locate this angle first at pi over three right here in quadrant one and I find that x value which is one half. Now whenever I take the reciprocal I'm really just flipping the fraction so here that ends up giving me two over one. Now two over one is just two and that's my final answer here that the secant of pi over three is equal to two. Thanks for watching and let me know if you have questions.

Evaluate each expression. sec ⁡ ( π 3 ) \sec\left(\frac{\pi}{3}\right) sec ( 3 π )

3 2 \frac{\sqrt3}{2} 2 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

2 3 2 \frac{2\sqrt3}{2} 2 2 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

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Multiple Choice

Evaluate each expression.
$\sec\left(\frac{\pi}{3}\right)$

$\frac12$

$2$

$\frac{\sqrt3}{2}$

$\frac{2\sqrt3}{2}$

Verified step by step guidance

Step 1: Understand the problem. The goal is to evaluate the secant function for the given angles. Recall that secant is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x).

Step 2: Start with the first term, sec(π/3). Use the unit circle to find cos(π/3). From trigonometric values, cos(π/3) = 1/2. Therefore, sec(π/3) = 1/(1/2).

Step 3: Next, evaluate sec(3π). Note that 3π is a multiple of π, and cosine values for multiples of π are either 1 or -1. Specifically, cos(3π) = -1. Thus, sec(3π) = 1/(-1).

Step 4: Combine the results from the previous steps. Substitute the evaluated values of sec(π/3) and sec(3π) into the expression to simplify further.

Step 5: Simplify the final expression to obtain the result. Ensure all fractions and radicals are simplified appropriately.