Question

use the trigonometric values for special angles for \\( \sin \theta \\) and the reciprocal identity \\( \csc \theta = \frac { 1 } { \sin \theta } \\) to calculate \\( \csc 45 ^ { \circ } \\). rationalize any denominators containing radicals that you encounter in the answer.
note: enter the exact, fully simplified answer.
\\( \csc 45 ^ { \circ } = \\)

Explanation:

Step1: Recall the value of $\sin45^{\circ}$

We know that $\sin45^{\circ}=\frac{\sqrt{2}}{2}$.

Step2: Use the reciprocal identity $\csc\theta=\frac{1}{\sin\theta}$

Substitute $\theta = 45^{\circ}$ into the identity. So, $\csc45^{\circ}=\frac{1}{\sin45^{\circ}}$.
Since $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, then $\csc45^{\circ}=\frac{1}{\frac{\sqrt{2}}{2}}$.
Using the rule $\frac{1}{\frac{a}{b}}=\frac{b}{a}$ ($a = \sqrt{2}$, $b = 2$), we get $\csc45^{\circ}=\frac{2}{\sqrt{2}}$.

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{2}$.
$\frac{2}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}}{2}$.
Simplify the fraction $\frac{2\sqrt{2}}{2}$ by canceling out the common factor of 2.

Answer:

$\sqrt{2}$