Question

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

Answer 1

`3/sqrt5`

Explanation:

Answer 2

Using right angle trigonometry, we can find out that `sec(theta)=(3sqrt(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(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 `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(theta)=1/cos(theta)=1/(b/c)=c/b`, and since we know `c` and `b`, we can solve for `sec(theta)`, which equals `(3sqrt(5))/5.`