3. if the angle $\\theta=\\frac{7\\pi}{4}$ is in standard position on the unit circle, which ordered pair…

3. if the angle $\\theta=\\frac{7\\pi}{4}$ is in standard position on the unit circle, which ordered pair does its terminal side pass through?\n$\\bigcirc(\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(-\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(-\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2})$\n6. ar ar

3. if the angle $\\theta=\\frac{7\\pi}{4}$ is in standard position on the unit circle, which ordered pair does its terminal side pass through?\n$\\bigcirc(\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(-\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(-\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2})$\n$\\bigcirc(\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2})$\n6. ar ar

Answer

Explanation:

Step1: Recall the unit - circle coordinates formula

For an angle (\theta) in standard position on the unit circle, the coordinates of the point on the terminal side are ((\cos\theta,\sin\theta)).

Step2: Find the reference angle

The angle (\theta=\frac{7\pi}{4}). The reference angle (\theta_{r}=2\pi - \frac{7\pi}{4}=\frac{\pi}{4}).

Step3: Determine the signs of (\cos\theta) and (\sin\theta)

Since (\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi), the angle (\frac{7\pi}{4}) is in the fourth quadrant. In the fourth quadrant, (\cos\theta>0) and (\sin\theta < 0).

Step4: Calculate (\cos\theta) and (\sin\theta)

We know that (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}) and (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}). So, (\cos\frac{7\pi}{4}=\cos(2\pi - \frac{\pi}{4})=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}) (using the identity (\cos(2\pi - \alpha)=\cos\alpha)) and (\sin\frac{7\pi}{4}=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}) (using the identity (\sin(2\pi - \alpha)=-\sin\alpha))

Answer:

(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right))