Question

haruto simplified the value below. csc(19π/4)=√2. which statement explains whether haruto is correct? haruto is correct because the angle is coterminal with 3π/4 and the reference angle is π/4. haruto is correct because the angle is coterminal with π/4, which is also the reference angle. haruto is not correct because the angle is coterminal with 3π/4, and csc(3π/4)= -1/√2=-√2/2. haruto is not correct because the angle is coterminal with π/4, and csc(π/4)=1/√2 = √2/2.

Explanation:

Step1: Find a coterminal angle

To find a coterminal angle of $\frac{19\pi}{4}$, subtract \(4\pi\) (since \(4\pi = \frac{16\pi}{4}\)). So, \(\frac{19\pi}{4}-4\pi=\frac{19\pi - 16\pi}{4}=\frac{3\pi}{4}\).

Step2: Recall the definition of cosecant

The cosecant function is defined as \(\csc\theta=\frac{1}{\sin\theta}\). For \(\theta = \frac{3\pi}{4}\), \(\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\), then \(\csc(\frac{3\pi}{4})=\frac{1}{\sin(\frac{3\pi}{4})}=\frac{1}{\frac{\sqrt{2}}{2}}=\sqrt{2}\). Also, the reference - angle of \(\frac{3\pi}{4}\) is \(\pi-\frac{3\pi}{4}=\frac{\pi}{4}\).

Answer:

Haruto is correct because the angle is coterminal with \(\frac{3\pi}{4}\) and the reference angle is \(\frac{\pi}{4}\).