Question

answer the following. give exact values for your answers.
(a) find an angle between 0 and 2π that is coterminal with (\frac{11pi}{3}).
(b) find an angle between 0° and 360° that is coterminal with - 240°

Explanation:

Step1: Recall coterminal - angle formula for radians

To find a coterminal angle of $\theta$ in radians, we use $\theta + 2k\pi$, where $k\in\mathbb{Z}$. Given $\theta=\frac{17\pi}{3}$, we want $0\leq\theta + 2k\pi\leq2\pi$.
Let's solve the inequality $0\leq\frac{17\pi}{3}+ 2k\pi\leq2\pi$. First, subtract $\frac{17\pi}{3}$ from all parts: $-\frac{17\pi}{3}\leq2k\pi\leq2\pi-\frac{17\pi}{3}=-\frac{11\pi}{3}$. Then divide by $2\pi$: $-\frac{17}{6}\leq k\leq-\frac{11}{6}$. Since $k$ is an integer, $k = - 2$.

Step2: Calculate the coterminal angle

Substitute $k=-2$ into $\theta + 2k\pi$. We have $\frac{17\pi}{3}+2\times(-2)\pi=\frac{17\pi}{3}-4\pi=\frac{17\pi - 12\pi}{3}=\frac{5\pi}{3}$.

Step3: Recall coterminal - angle formula for degrees

To find a coterminal angle of $\alpha$ in degrees, we use $\alpha + 360^{\circ}k$, where $k\in\mathbb{Z}$. Given $\alpha=-240^{\circ}$, we want $0^{\circ}\leq\alpha + 360^{\circ}k\leq360^{\circ}$.
Let's solve the inequality $0^{\circ}\leq - 240^{\circ}+360^{\circ}k\leq360^{\circ}$. Add $240^{\circ}$ to all parts: $240^{\circ}\leq360^{\circ}k\leq600^{\circ}$. Then divide by $360^{\circ}$: $\frac{2}{3}\leq k\leq\frac{5}{3}$. Since $k$ is an integer, $k = 1$.

Step4: Calculate the coterminal angle

Substitute $k = 1$ into $\alpha+360^{\circ}k$. We get $-240^{\circ}+360^{\circ}\times1 = 120^{\circ}$.

Answer:

(a) $\frac{5\pi}{3}$
(b) $120^{\circ}$