Find the exact values of the remaining trigonometric functions of θ satisfying the given conditions

Question

Find the exact values of the remaining trigonometric functions of θ satisfying the given conditions. (If an answer is undefined, enter UNDEFINED.)

cos θ = 15/17

tan θ < 0

,

sin θ	=
tan θ	=
csc θ	=
sec θ	=
cot θ	=

Answer 1

Hello Natalie. Given that cosine is positive but tangent is negative, the angle you're working with must be in the 4th quadrant ( per "All Students Take Calculus", if you're familiar with that memory aid!). So if we draw a diagram and set up a triangle in the 4th quadrant with a reference angle of "theta", we can then label the adjacent side of the triangle ( i.e. the side lying on the positive x-axis) as 15, and the hypotenuse of the right triangle as 17. By Pythagorean Theorem, we can determine that the side opposite angle "theta" ( i.e. the vertical side going down from the x-axis) is 8 units in length.

Now we are ready to determine the ratios for the remaining 5 trig functions, keeping in mind which ratios will be positive in the 4th quadrant and which ones will be negative. So here we go:

sin = -8/17;

tan = -8/15;

cot = -15/8

sec = 17/15;

csc = -17/8