Math / Trigonometry

Question(a) Sketch $\theta=\frac{3 \pi}{4}$ in standard position.
Then sketch an angle between $-2 \pi$ and 0 that is coterminal with $\theta$ . (b) Find the measure of the coterminal angle. Write your answer in terms of $\pi$ . Your answer should b between $-2 \pi$ and 0 . $\square$ radians

Studdy Solution

STEP 1

1. The angle $\theta = \frac{3\pi}{4}$ is given in standard position.
2. A coterminal angle is an angle that shares the same terminal side as another angle.
3. We need to find a coterminal angle for $\theta$ that lies between $-2\pi$ and $0$ .

STEP 2

1. Sketch the angle $\theta = \frac{3\pi}{4}$ in standard position.
2. Determine a coterminal angle between $-2\pi$ and $0$ .
3. Calculate the measure of the coterminal angle.

STEP 3

To sketch $\theta = \frac{3\pi}{4}$ in standard position, start from the positive x-axis and rotate counterclockwise. $\frac{3\pi}{4}$ is in the second quadrant, as it is more than $\frac{\pi}{2}$ but less than $\pi$ .

STEP 4

To find a coterminal angle between $-2\pi$ and $0$ , subtract $2\pi$ from $\theta$ . This is because subtracting $2\pi$ will give us an equivalent angle that is negative.

STEP 5

Calculate the coterminal angle: $\theta_{\text{coterminal}} = \frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}$

STEP 6

The measure of the coterminal angle is $-\frac{5\pi}{4}$ , which lies between $-2\pi$ and $0$ .
The coterminal angle is:
$\boxed{-\frac{5\pi}{4}}$

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Question(a) Sketch $\theta=\frac{3 \pi}{4}$ in standard position.
Then sketch an angle between $-2 \pi$ and 0 that is coterminal with $\theta$ . (b) Find the measure of the coterminal angle. Write your answer in terms of $\pi$ . Your answer should b between $-2 \pi$ and 0 . $\square$ radians

Studdy Solution

STEP 1

1. The angle $\theta = \frac{3\pi}{4}$ is given in standard position.
2. A coterminal angle is an angle that shares the same terminal side as another angle.
3. We need to find a coterminal angle for $\theta$ that lies between $-2\pi$ and $0$ .

STEP 2

1. Sketch the angle $\theta = \frac{3\pi}{4}$ in standard position.
2. Determine a coterminal angle between $-2\pi$ and $0$ .
3. Calculate the measure of the coterminal angle.

STEP 3

To sketch $\theta = \frac{3\pi}{4}$ in standard position, start from the positive x-axis and rotate counterclockwise. $\frac{3\pi}{4}$ is in the second quadrant, as it is more than $\frac{\pi}{2}$ but less than $\pi$ .

STEP 4

To find a coterminal angle between $-2\pi$ and $0$ , subtract $2\pi$ from $\theta$ . This is because subtracting $2\pi$ will give us an equivalent angle that is negative.

STEP 5

Calculate the coterminal angle: $\theta_{\text{coterminal}} = \frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}$

STEP 6

The measure of the coterminal angle is $-\frac{5\pi}{4}$ , which lies between $-2\pi$ and $0$ .
The coterminal angle is:
$\boxed{-\frac{5\pi}{4}}$

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Studdy Solution

STEP 1

1. The angle θ=3π4\theta = \frac{3\pi}{4}θ=43π​ is given in standard position.2. A coterminal angle is an angle that shares the same terminal side as another angle.3. We need to find a coterminal angle for θ\thetaθ that lies between −2π-2\pi−2π and 000.

STEP 2

1. Sketch the angle θ=3π4\theta = \frac{3\pi}{4}θ=43π​ in standard position.2. Determine a coterminal angle between −2π-2\pi−2π and 000.3. Calculate the measure of the coterminal angle.

STEP 3

To sketch θ=3π4\theta = \frac{3\pi}{4}θ=43π​ in standard position, start from the positive x-axis and rotate counterclockwise. 3π4\frac{3\pi}{4}43π​ is in the second quadrant, as it is more than π2\frac{\pi}{2}2π​ but less than π\piπ.

STEP 4

To find a coterminal angle between −2π-2\pi−2π and 000, subtract 2π2\pi2π from θ\thetaθ. This is because subtracting 2π2\pi2π will give us an equivalent angle that is negative.

STEP 5

Calculate the coterminal angle: θcoterminal=3π4−2π=3π4−8π4=−5π4\theta_{\text{coterminal}} = \frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}θcoterminal​=43π​−2π=43π​−48π​=−45π​

STEP 6

The measure of the coterminal angle is −5π4-\frac{5\pi}{4}−45π​, which lies between −2π-2\pi−2π and 000. The coterminal angle is:−5π4 \boxed{-\frac{5\pi}{4}} −45π​​

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1. The angle $\theta = \frac{3\pi}{4}$ is given in standard position.
2. A coterminal angle is an angle that shares the same terminal side as another angle.
3. We need to find a coterminal angle for $\theta$ that lies between $-2\pi$ and $0$ .

1. Sketch the angle $\theta = \frac{3\pi}{4}$ in standard position.
2. Determine a coterminal angle between $-2\pi$ and $0$ .
3. Calculate the measure of the coterminal angle.

To sketch $\theta = \frac{3\pi}{4}$ in standard position, start from the positive x-axis and rotate counterclockwise. $\frac{3\pi}{4}$ is in the second quadrant, as it is more than $\frac{\pi}{2}$ but less than $\pi$ .

To find a coterminal angle between $-2\pi$ and $0$ , subtract $2\pi$ from $\theta$ . This is because subtracting $2\pi$ will give us an equivalent angle that is negative.

Calculate the coterminal angle: $\theta_{\text{coterminal}} = \frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}$

The measure of the coterminal angle is $-\frac{5\pi}{4}$ , which lies between $-2\pi$ and $0$ .
The coterminal angle is:
$\boxed{-\frac{5\pi}{4}}$