Question

determine in which of the four quadrants angle $\theta$ is located given $cos(\theta)>0$ and $csc(\theta)>0$

Explanation:

Step1: Recall trigonometric identities

Recall that $\csc(\theta)=\frac{1}{\sin(\theta)}$. Since $\csc(\theta)> 0$, then $\sin(\theta)>0$ (because for $\csc(\theta)=\frac{1}{\sin(\theta)}$ to be positive, $\sin(\theta)$ must be positive as the reciprocal of a positive number is positive).

Step2: Recall signs of trig functions in quadrants

In the unit - circle, $\cos(\theta)$ represents the $x$-coordinate of a point on the unit - circle corresponding to the angle $\theta$, and $\sin(\theta)$ represents the $y$-coordinate.
We know that $\cos(\theta)>0$ when the $x$-coordinate is positive (this occurs in Quadrants I and IV).
We also know that $\sin(\theta)>0$ when the $y$-coordinate is positive (this occurs in Quadrants I and II).

Step3: Find the common quadrant

The quadrant where both $\cos(\theta)>0$ and $\sin(\theta)>0$ is Quadrant I.

Answer:

Quadrant I