Questions: For each angle below, determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle θ̅. [NOTE 1: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.] [NOTE 2: You can enter π as 'pi' in your answers.] (a) θ=8π/3 is found in quadrant and θ̅= DNE (b) θ=3π/4 is found in quadrant and θ̅= (c) θ=-1π/6 is found in quadrant and θ̅= (d) θ=4 is found in quadrant and θ̅=

For each angle below, determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle θ̅.
[NOTE 1: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.]
[NOTE 2: You can enter π as 'pi' in your answers.]
(a) θ=8π/3 is found in quadrant and θ̅= DNE
(b) θ=3π/4 is found in quadrant and θ̅=
(c) θ=-1π/6 is found in quadrant and θ̅=
(d) θ=4 is found in quadrant and θ̅=

Transcript text: For each angle below, determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle $\bar{\theta}$. [NOTE 1: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.] [NOTE 2: You can enter $\pi$ as 'pi' in your answers.] (a) $\theta=\frac{8 \pi}{3}$ is found in quadrant $\square$ and $\bar{\theta}=$ DNE $\square$ (b) $\theta=\frac{3 \pi}{4}$ is found in quadrant $\square$ and $\bar{\theta}=$ $\square$ (c) $\theta=\frac{-1 \pi}{6}$ is found in quadrant $\square$ and $\bar{\theta}=$ $\square$ (d) $\theta=4$ is found in quadrant $\square$ and $\bar{\theta}=$ $\square$

Solution

Solution Steps

Solution Approach

To determine the quadrant and reference angle for each given angle:

Convert the angle to a standard position by finding its equivalent between $0$ and $2\pi$ using modulo $2\pi$.
Determine the quadrant by checking the range in which the angle lies:
- Quadrant I: $0 < \theta < \frac{\pi}{2}$
- Quadrant II: $\frac{\pi}{2} < \theta < \pi$
- Quadrant III: $\pi < \theta < \frac{3\pi}{2}$
- Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$
Calculate the reference angle $\bar{\theta}$ based on the quadrant:
- Quadrant I: $\bar{\theta} = \theta$
- Quadrant II: $\bar{\theta} = \pi - \theta$
- Quadrant III: $\bar{\theta} = \theta - \pi$
- Quadrant IV: $\bar{\theta} = 2\pi - \theta$

Step 1: Normalize the Angles

For each angle, we first normalize it to be within the range $0$ to $2\pi$:

For $\theta = \frac{8\pi}{3}$: \[ \frac{8\pi}{3} \mod 2\pi = \frac{8\pi}{3} - 2\pi = \frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3} \]
For $\theta = \frac{3\pi}{4}$, it is already in the range.
For $\theta = -\frac{\pi}{6}$: \[ -\frac{\pi}{6} + 2\pi = -\frac{\pi}{6} + \frac{12\pi}{6} = \frac{11\pi}{6} \]
For $\theta = 4$, we convert it to radians: \[ 4 \text{ radians} \text{ (already in range)} \]

Step 2: Determine the Quadrants

Next, we determine the quadrant for each normalized angle:

For $\frac{2\pi}{3}$: This angle is in Quadrant II.
For $\frac{3\pi}{4}$: This angle is also in Quadrant II.
For $\frac{11\pi}{6}$: This angle is in Quadrant IV.
For $4$ radians: This angle is in Quadrant III.

Step 3: Calculate the Reference Angles

Now, we calculate the reference angles for each:

For $\frac{2\pi}{3}$ (Quadrant II): \[ \bar{\theta} = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3} \approx 1.0472 \]
For $\frac{3\pi}{4}$ (Quadrant II): \[ \bar{\theta} = \pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4} \approx 0.7854 \]
For $\frac{11\pi}{6}$ (Quadrant IV): \[ \bar{\theta} = 2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6} \approx 0.5236 \]
For $4$ radians (Quadrant III): \[ \bar{\theta} = 4 - \pi \approx 4 - 3.1416 \approx 0.8584 \]

Final Answer

The results for each angle are as follows:

(a) $\theta = \frac{8\pi}{3}$ is found in quadrant $2$ and $\bar{\theta} = \frac{\pi}{3}$.
(b) $\theta = \frac{3\pi}{4}$ is found in quadrant $2$ and $\bar{\theta} = \frac{\pi}{4}$.
(c) $\theta = -\frac{\pi}{6}$ is found in quadrant $4$ and $\bar{\theta} = \frac{\pi}{6}$.
(d) $\theta = 4$ is found in quadrant $3$ and $\bar{\theta} \approx 0.8584$.

Thus, the answers are: