Question

use the given triangles to evaluate the following expression. if necessary, express the value without a square root in the denominator by rationalizing the denominator. csc(π/4)

Explanation:

Step1: Recall the definition of cosecant

The cosecant function is defined as $\csc\theta=\frac{1}{\sin\theta}$. So, $\csc(\frac{\pi}{4})=\frac{1}{\sin(\frac{\pi}{4})}$.

Step2: Find the value of $\sin(\frac{\pi}{4})$

In a $45 - 45-90$ triangle, for an angle of $45^{\circ}$ (or $\frac{\pi}{4}$ radians), $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\theta = 45^{\circ}$, if the opposite side and adjacent side are of length $1$ and the hypotenuse is $\sqrt{2}$, then $\sin(\frac{\pi}{4})=\frac{1}{\sqrt{2}}$.

Step3: Calculate $\csc(\frac{\pi}{4})$

Substitute $\sin(\frac{\pi}{4})=\frac{1}{\sqrt{2}}$ into the formula $\csc(\frac{\pi}{4})=\frac{1}{\sin(\frac{\pi}{4})}$. We get $\csc(\frac{\pi}{4})=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2}$.

Answer:

$\sqrt{2}$