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

2 Answers
Jun 4, 2018

3/sqrt5

Explanation:

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Jun 4, 2018

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:

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(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.