If sin theta = 8/17 and cot theta < 0, what is sec theta?

Step-by-step explanation:

By definition, and . Since since is negative, must also be negative, and since is positive, we must be in Quadrant II.

In a right triangle, the sine of an angle is equal to its opposite side divided by the hypotenuse. The cosine of an angle in a right triangle is equal to its adjacent side divided by the hypotenuse. Therefore, we can draw a right triangle in Quadrant II, where the opposite side to angle theta is 8 and the hypotenuse of the triangle is 17.

To find the remaining leg, use to the Pythagorean Theorem, where , where is the hypotenuse, or longest side, of the right triangle and and are the two legs of the right triangle.

Solving, we get:

Since all values of cosine theta are negative in Quadrant II, all values of secant theta must also be negative in Quadrant II.

Thus, we have: