Haruto simplified the value below. $$ \csc \left(\frac{19 \pi}{4}\right)=\sqrt{2} $$ Which statement explains whether Haruto is correct?

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UpStudy AI · Accepted Answer

1. First, note that the cosecant function is the reciprocal of the sine function, so  
   $$
   \csc\left(\frac{19\pi}{4}\right) = \frac{1}{\sin\left(\frac{19\pi}{4}\right)}.
   $$
   
2. Reduce the angle $$\frac{19\pi}{4}$$ modulo $$2\pi$$. Since $$2\pi = \frac{8\pi}{4}$$, subtract multiples of $$\frac{8\pi}{4}$$:
   $$
   \frac{19\pi}{4} - \frac{16\pi}{4} = \frac{3\pi}{4}.
   $$
   Thus,
   $$
   \sin\left(\frac{19\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right).
   $$

3. Evaluate $$\sin\left(\frac{3\pi}{4}\right)$$. The sine of $$\frac{3\pi}{4}$$ is known to be:
   $$
   \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}.
   $$

4. Substitute the sine value back into the reciprocal:
   $$
   \csc\left(\frac{19\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}}.
   $$

5. Simplify $$\frac{2}{\sqrt{2}}$$ by rationalizing:
   $$
   \frac{2}{\sqrt{2}} = \sqrt{2}.
   $$

Since the final result is $$\sqrt{2}$$, Haruto is correct.
Haruto is correct because $$\csc\left(\frac{19\pi}{4}\right) = \sqrt{2}$$.

Haruto simplified the value below.

Which statement explains whether Haruto is correct?

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Haruto simplified the value below. Which statement explains whether Haruto is correct?