Question

Determined the quadrant for an angle with the following characteristics: cotθ>0 and cscθ>0.

A) Quadrant I
B) Quadrant II
C) Quadrant III
D) Quadrant IV

Answer 1

The quadrant for an angle with cotθ>0 and cscθ>0 is Quadrant I.

Step-by-step explanation:

To determine the quadrant for an angle with the characteristics cotθ>0 and cscθ>0, we need to analyze the signs of the trigonometric functions in each quadrant. In Quadrant I, both cotangent and cosecant are positive, so it satisfies the given conditions. Therefore, the answer is Quadrant I.

Answer 2

`\csc\theta=\frac1{\sin\theta}>0\implies\sin\theta>0`

This happens whenever the terminal point of
`\theta` lies in either the first or second quadrants.

Meanwhile,


`\cot\theta=\frac1{\tan\theta}=(\cos\theta)/(\sin\theta)>0\implies\cos\theta>0`

since you already know sine is positive. Cosine is positive when the angle lies in the first or fourth quadrant.

Sine and cosine are both positive only when
`\theta` is the first quadrant, which means this angle's terminal point lies in the first quadrant (A).