Question

in exercises 1 and 2, identify the one angle that is not coterminal with all the others.

1. ( 150^{circ}, 510^{circ}, -210^{circ}, 450^{circ}, 870^{circ} )
2. ( \frac{5pi}{3}, -\frac{5pi}{3}, \frac{11pi}{3}, -\frac{7pi}{3}, \frac{365pi}{3} )

Explanation:

Step1: Find coterminal angles for degrees

To find coterminal angles for degrees, we use the formula \(\theta + n\times360^{\circ}\), where \(n\in\mathbb{Z}\).
For \(150^{\circ}\):

• \(150^{\circ}+ 360^{\circ}=510^{\circ}\)
• \(150^{\circ}-360^{\circ}=- 210^{\circ}\)
• \(150^{\circ}+2\times360^{\circ}=150^{\circ}+720^{\circ}=870^{\circ}\)
• \(150^{\circ}+360^{\circ}\times1 = 510^{\circ}\), \(150^{\circ}-360^{\circ}\times1=-210^{\circ}\), \(150^{\circ}+360^{\circ}\times2 = 870^{\circ}\)
• For \(450^{\circ}\), \(450^{\circ}-360^{\circ}=90^{\circ}

eq150^{\circ}\)

Step2: Find coterminal angles for radians

To find coterminal angles for radians, we use the formula \(\alpha + 2n\pi\), where \(n\in\mathbb{Z}\).
For \(\frac{5\pi}{3}\):

• \(\frac{5\pi}{3}-2\pi=\frac{5\pi - 6\pi}{3}=-\frac{\pi}{3}

eq-\frac{5\pi}{3}\)

• \(\frac{5\pi}{3}+2\pi=\frac{5\pi + 6\pi}{3}=\frac{11\pi}{3}\)
• \(\frac{5\pi}{3}-4\pi=\frac{5\pi-12\pi}{3}=-\frac{7\pi}{3}\)
• \(\frac{5\pi}{3}+24\pi=\frac{5\pi + 72\pi}{3}=\frac{365\pi}{3}\)

Answer:

1. \(450^{\circ}\)
2. \(-\frac{5\pi}{3}\)