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

Jun 4, 2018

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

Explanation:

Jun 4, 2018

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

Explanation:

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

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

We are told that $\sin \left(\theta\right) = \frac{2}{3}$, 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 $\frac{a}{c}$. So $a = 2$ and $c = 3$. Then we can use the Pythagorean Theorem to figure out that side $b$ equals $\sqrt{{c}^{2} - {a}^{2}} = \sqrt{9 - 4} = \sqrt{5}$. We know that $\sec \left(\theta\right) = \frac{1}{\cos} \left(\theta\right) = \frac{1}{\frac{b}{c}} = \frac{c}{b}$, and since we know $c$ and $b$, we can solve for $\sec \left(\theta\right)$, which equals $\frac{3 \sqrt{5}}{5.}$