Let θ be an angle in quadrant II such that cosθ=−1/4. Find the exact values of cscθ and cotθ .

In the second quadrant, if the cos(θ) = -1/4, then a right triangle can be drawn with adjacent side = -1 and hypotenuse = 4.

To find the opposite side:  c² = a² + b²   ---> 4² = 1² + b²    --->   b = √15  (this is positive because it is a y-value in the second quadrant.

The value of sin(θ) = √15/4     and the value of tan(θ) = -√15

csc(θ) = 1/sin(θ)  =  1/(√15/4)  =  4/√15  =  4√15/15

cot(θ) = 1/tan(θ)  =  1/(1/-√15)  =  -√15