Find the terminal point on the unit circle determined by 7pi/4 radians. Use exact values, not decimal approximations.
The Correct Answer and Explanation is:
To find the terminal point on the unit circle determined by 7π4\frac{7\pi}{4}47π radians, we need to determine both the reference angle and the coordinates of the terminal point.
Step 1: Find the reference angle.
A reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. Since 7π4\frac{7\pi}{4}47π is greater than 2π2\pi2π (a full revolution), we need to subtract 2π2\pi2π to get the corresponding angle between 000 and 2π2\pi2π. 7π4−2π=7π4−8π4=−π4\frac{7\pi}{4} – 2\pi = \frac{7\pi}{4} – \frac{8\pi}{4} = -\frac{\pi}{4}47π−2π=47π−48π=−4π
This tells us that 7π4\frac{7\pi}{4}47π is coterminal with −π4-\frac{\pi}{4}−4π, which is the reference angle.
Step 2: Identify the quadrant.
The angle 7π4\frac{7\pi}{4}47π lies in the fourth quadrant. This is because:
- 7π4\frac{7\pi}{4}47π is between 2π2\pi2π (which is 8π4\frac{8\pi}{4}48π) and 2π+π42\pi + \frac{\pi}{4}2π+4π, which is the fourth quadrant of the unit circle.
Step 3: Use the reference angle to determine the coordinates.
The reference angle is π4\frac{\pi}{4}4π, and the coordinates for an angle of π4\frac{\pi}{4}4π on the unit circle are: (cos(π4),sin(π4))=(22,22)\left( \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right) \right) = \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)(cos(4π),sin(4π))=(22,22)
Since 7π4\frac{7\pi}{4}47π lies in the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. Therefore, the coordinates of the terminal point are: (22,−22)\left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right)(22,−22)
Final Answer:
The terminal point on the unit circle determined by 7π4\frac{7\pi}{4}47π radians is: (22,−22)\left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right)(22,−22)
