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 3. Now remember, the secant is the reciprocal of cosine. So here I can take 1 over my cosine value, which I know is x on the unit circle. So I can locate this angle first at pi over 3 right here in quadrant 1 and I find that x value which is 1 half. Now whenever I take the reciprocal, I'm really just flipping the fraction, 2 over my final answer here, that the secant of pi over 3 is equal to 2. 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">

3. Unit Circle

Reciprocal Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Reciprocal 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

Understand that secant, $ \sec(\theta) $, is the reciprocal of cosine, $ \cos(\theta) $. Therefore, $ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} $.

Recall the value of $ \cos\left(\frac{\pi}{3}\right) $. From the unit circle or trigonometric tables, $ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $.

Substitute the value of $ \cos\left(\frac{\pi}{3}\right) $ into the secant expression: $ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} $.

Simplify the expression $ \frac{1}{\frac{1}{2}} $ to find the value of $ \sec\left(\frac{\pi}{3}\right) $.

Conclude that the simplified value of $ \sec\left(\frac{\pi}{3}\right) $ is 2.