Question

if cot theta = -24/7 and cos theta < 0, name the quad in which theta lies, then find the exact value of csc theta

Answer 1

Cot θ = -24/7 and cos θ < 0

"cos θ < 0" means cosine is negative

Remember osmething like "ALL students Take Calculus"?

all is positive in quadrant 1
sine/cosecant is positive only in quadrant 2
tangent/cotangent is positive only in quadrant 3
cosine/secant is positive only in quadrant 4

since cot(angent) is the reciprocal of tangent, tangent must also be negative.

tangent is negative in quadrants II and IV
cosine is negative in quadrants II and III

therefore, θ must lie in quadrant II.

remember the tan θ = y / x. it is the y-coordinate divided by the x-coodinate. the y-coordinate would be the height of this 90 degree triangle that forms. the x-coordinate would be the bottom leg of the 90 degree triangle

then cot θ = x / y. therefore, cot θ = -24 / 7 means x = -24 and y = 7. we draw a triangle with these dimensions.

csc θ = r / y = hypotenuse over opposite (when you view the reference angle labeled as the one to base the hypotenuse/oppoiste on) because it is the reciprocal of sine

the hypotenuse can be found w/ pythagorean theorem

hypotenuse = √[(-24)² + 7²] = 25

therefore,
cot θ = 25/7