coordinate plane trig

Question

First, determine the quadrant for ; then find , , and ; and finally, give all six trigonometric ratios for  given the following information:

csc(θ)=−19/4 and cos(θ)<0

Answer 1

cosecant (csc) is negative when sine is negative [csc(θ) = 1/sin(θ)] in other words, since sin(θ) = opposite/hypotenuse, then csc(θ) = hypotenuse/opposite. So for csc(θ) = - 19/4 we can say the hypotenuse is 19 and the opposite side is -4. Also we are given the fact that cosine is less than zero (negative). A diagram looks like this:

where the top diagram shows the quadrants and their associated sine, cosine, and tangent values and the bottom shows this case. You can see that the location is in Quadrant III. Using the Pythagorean Theorem we can solve for the adjacent side;

a² + b² = c²

a² + (-4)² = 19²

a² + 16 = 361

a² = 361 - 16 = 345

a = √345 but since this is in Q3, then the adjacent side is -√345

sin(θ) = opposite/hypotenuse = - 4/19

cos(θ) = adjacent/hypotenuse = - √345/19

tan(θ) = opposite/adjacent = (-4)/(-√345) = 4/√345 = 4√345/345

csc(θ) = hypotenuse/opposite = - 19/4

sec(θ) = hypotenuse/adjacent = 19/(-√345) = - 19√345/345

cot(θ) = adjacent/opposite = (-√345)/(-4) = √345/4