If sin theta=2/3 with theta in quadrant 1, find sec theta?

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If sin theta=2/3 with theta in quadrant 1, find sec theta?

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Answer 1 · 2018-06-04T20:55:23

$\frac{3}{\sqrt{5}}$ $3/sqrt5$

Explanation:

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Answer 2 · 2018-06-04T20:57:39

Using right angle trigonometry, we can find out that $sec (θ) = \frac{3 \sqrt{5}}{5}$ $sec(theta)=(3sqrt(5))/5$ .

Explanation:

To solve this problem, we can use a right triangle. Here is what mine looks like:

![d2gne97vdumgn3.cloudfront.net](useruploads.socratic.org)

(We know the triangle is in this orientation because theta is in the first quadrant).

We are told that $sin (θ) = \frac{2}{3}$ $sin(theta)=2/3$ , so by using the very helpful tool SOHCAHTOA we know that sine is the opposite of the triangle over the hypotenuse of the triangle, which here is $\frac{a}{c}$ $a/c$ . So $a = 2$ $a=2$ and $c = 3$ $c=3$ . Then we can use the Pythagorean Theorem to figure out that side $b$ $b$ equals $\sqrt{c^{2} - a^{2}} = \sqrt{9 - 4} = \sqrt{5}$ $sqrt(c^2-a^2)=sqrt(9-4)=sqrt(5)$ . We know that $sec (θ) = \frac{1}{cos (θ)} = \frac{1}{\frac{b}{c}} = \frac{c}{b}$ $sec(theta)=1/cos(theta)=1/(b/c)=c/b$ , and since we know $c$ $c$ and $b$ $b$ , we can solve for $sec (θ)$ $sec(theta)$ , which equals $\frac{3 \sqrt{5}}{5} .$ $(3sqrt(5))/5.$

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If sin theta=2/3 with theta in quadrant 1, find sec theta?

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