Question 398188
Suppose sec alpha = -7/4 and that the terminal side of alpha is in quadrant III. find sin alpha and tan alpha.

To start, it might help to visualize a picture of the right triangle in question.
Picture a circle with radius 7 and moving the terminal side to the reference angle (alpha) in quadrant III until the cos (horizonal)leg is equal to 4. You now have a right triangle with hypotenuse =7, and the adjacent leg=4. As you can see, both the horizontal (cos) and vertical (sin) legs are negative with respect to the origin.

Use the pythagorean theorem to find the vertical leg, call it x
x=sqrt(7^2-4^2)=sqrt(33)
sin alpha=(opposite/hypotenuse)=sqrt(33)/7
tan alpha = (opposite/adjacent)=sqrt(33)/4

In summary, this is what you have.

Sec alpha = -7/4 (given)
cos alpha = -4/7
sin alpha = sqrt(33)/7
tan alpha = sqrt(33)/4