Question

evaluate without a calculator: csc 225° type + or -

Explanation:

Step1: Determine the quadrant

$225^{\circ}$ is in the third quadrant. In the third quadrant, $\sin\theta<0$, and since $\csc\theta=\frac{1}{\sin\theta}$, $\csc225^{\circ}<0$.

Step2: Find the reference angle

The reference angle for $225^{\circ}$ is $225^{\circ}- 180^{\circ}=45^{\circ}$.

Step3: Evaluate using the reference angle

We know that $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, so $\sin225^{\circ}=-\sin45^{\circ}=-\frac{\sqrt{2}}{2}$. Then $\csc225^{\circ}=\frac{1}{\sin225^{\circ}}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$.

Answer:

• $\sqrt{2}$