Sketch the angle with the given measure, and its reference triangle. Then find the required ratio?

Question

14. csc 225° =
15. cot -150° =

Answer 1

(14).

csc(225^o)

The angle 225^o lies in third quadrant and the reference angle is

θ' = 180^o + 45^o = 45^o.

The cosecant is negative in Quadrant III, so

csc(225^o) = (-) csc(45) = - (√2)/1.

We know that csc(θ) = r/y, therefore r = √2 and y = -1

Find the x-value by using formula: r² = x² + y².

2 = x² + (-1)² ⇒ x = ± 1.

The point lies in the third quadrant x < 0 and y < 0.

Therefore, point (x, y) = (-1, -1).

Graph:

Answer 2

15).

cot(-150^o)

Because - 150^o + 360^o = 210^o, it follows that -150^o is coterminal  with  the third-quadrant angle 210^o. So the reference angle is

θ' = 210^o - 180^o = 30^o.

The cotangent is positive in Quadrant III, so

cot(-150^o) = (+) cot(30) = √3/1.

We know that csc(θ) = x/y, therefore x = √3 and y = 1.

Find the r-value by using formula: r = √(x² + y²).

r = √(x² + y²) = r = √[(√3)² + (1)²] = = √4 = 2.

The point lies in the third quadrant x < 0 and y < 0.

Therefore, point (x, y) = (-√3, -1).

Graph: