Question

drag the tiles to the correct boxes to complete the pairs. match the angle measurements in radians with equivalent measurements less than or equal to 360°. \\(\frac{23\pi}{4}\\) \\(\frac{18\pi}{5}\\) \\(\frac{22\pi}{9}\\) \\(\frac{19\pi}{3}\\) 60° 288° 315° 80°

Explanation:

Step1: Recall the conversion formula

To convert radians to degrees, we use the formula \( \text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi} \). Also, if the angle in radians is more than \( 2\pi \) (or \( 360^\circ \)), we can subtract multiples of \( 2\pi \) (or \( 360^\circ \)) to get the coterminal angle within \( 0^\circ \) to \( 360^\circ \).

Step2: Convert \( \frac{23\pi}{4} \) to degrees

First, find the coterminal angle. \( \frac{23\pi}{4} - 2\times2\pi=\frac{23\pi}{4}- \frac{16\pi}{4}=\frac{7\pi}{4} \). Now convert \( \frac{7\pi}{4} \) to degrees: \( \frac{7\pi}{4}\times\frac{180^\circ}{\pi}= 315^\circ \).

Step3: Convert \( \frac{18\pi}{5} \) to degrees

Find the coterminal angle. \( \frac{18\pi}{5}- 2\pi=\frac{18\pi}{5}-\frac{10\pi}{5}=\frac{8\pi}{5} \). Convert \( \frac{8\pi}{5} \) to degrees: \( \frac{8\pi}{5}\times\frac{180^\circ}{\pi}= 288^\circ \).

Step4: Convert \( \frac{22\pi}{9} \) to degrees

Find the coterminal angle. \( \frac{22\pi}{9}- 2\pi=\frac{22\pi}{9}-\frac{18\pi}{9}=\frac{4\pi}{9} \)? Wait, no, wait. Wait, \( 2\pi=\frac{18\pi}{9} \), \( \frac{22\pi}{9}-\frac{18\pi}{9}=\frac{4\pi}{9} \)? No, that's not right. Wait, maybe I made a mistake. Wait, \( \frac{22\pi}{9} \) is less than \( 3\pi \) (since \( 3\pi=\frac{27\pi}{9} \)), but to get within \( 0 - 360^\circ \), we can also just convert directly. Wait, \( \frac{22\pi}{9}\times\frac{180^\circ}{\pi}= 440^\circ \), then subtract \( 360^\circ \), \( 440 - 360 = 80^\circ \). Wait, let's check: \( \frac{22\pi}{9}\times\frac{180}{\pi}=22\times20 = 440^\circ \), \( 440 - 360 = 80^\circ \). So \( \frac{22\pi}{9} \) is coterminal with \( 80^\circ \).

Step5: Convert \( \frac{19\pi}{3} \) to degrees

Find the coterminal angle. \( \frac{19\pi}{3}- 3\times2\pi=\frac{19\pi}{3}-\frac{18\pi}{3}=\frac{\pi}{3} \). Convert \( \frac{\pi}{3} \) to degrees: \( \frac{\pi}{3}\times\frac{180^\circ}{\pi}= 60^\circ \).

Answer:

• \( 60^\circ \) pairs with \( \frac{19\pi}{3} \)
• \( 288^\circ \) pairs with \( \frac{18\pi}{5} \)
• \( 315^\circ \) pairs with \( \frac{23\pi}{4} \)
• \( 80^\circ \) pairs with \( \frac{22\pi}{9} \)