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Trigonometry : Angle Applications

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Example Questions

Example Question #1 : Angle Applications

What is the formula for angular velocity?

Possible Answers:

$\omega = \frac{r}{l_A}$

$\omega = \frac{\theta}{l_A}$

$\omega = \frac{\theta}{t}$

$\omega = \frac{\theta}{r}$

Correct answer:

$\omega = \frac{\theta}{t}$

Explanation:

Angular velocity is a measure of the time it takes to travel over a certain arc that is formed by a central angle. $\omega$ is usually used to represent the angular velocity, $\theta$ is the measure of the central angle, and is the time it takes to travel from point A to B. This formula looks similar to the measure of an arc, $\theta = \frac{r}{l_A}$ , but we are considering the velocity in terms of the measure of the angle over the amount of time it took to cover that distance.

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Example Question #2 : Angle Applications

What is the angular velocity of an angle whose measure is $\frac{5\pi}{4}$ when it took a point 6 seconds to rotate through the angle.

Possible Answers:

$\omega = \frac{10\pi}{24}$ radians/seconds

$\omega = \frac{5\pi}{20}$ radians/seconds

$\omega = \frac{5\pi}{24}$ radians/seconds

$\omega = 5\pi$ radians/seconds

Correct answer:

$\omega = \frac{5\pi}{24}$ radians/seconds

Explanation:

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = \frac{5\pi}{4}$ and t = 6 seconds.

$\omega = \frac{\theta}{t}$

$\omega = (\frac{5\pi}{4})(\frac{1}{6})$

$\omega = \frac{5\pi}{24}$

So our answer is $\omega = \frac{5\pi}{24}$ radians/seconds

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Example Question #3 : Angle Applications

True or False: The angular velocity is a measure of an angle of rotation over time.

Possible Answers:

False

True

Correct answer:

True

Explanation:

consider the figure below. Think of line A being connected to a hinge where $\theta$ is. We are measuring the amount of time it takes for line A to rotate and align with line B. This is a measure of angle rotation over time.

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Example Question #4 : Angle Applications

What is the angular velocity of an angle whose measure is $\frac{\pi}{3}$ when it took a point 10 seconds to rotate through the angle.

Possible Answers:

$\omega = \frac{8\pi}{30}$ radians/seconds

$\omega = 10\pi$ radians/seconds

$\omega = \frac{\pi}{30}$ radians/seconds

$\omega = 30\pi$ radians/seconds

Correct answer:

$\omega = \frac{\pi}{30}$ radians/seconds

Explanation:

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = \frac{\pi}{3}$ and t = 10 seconds.

$\omega = \frac{\theta}{t}$

$\omega = (\frac{\pi}{3})(\frac{1}{10})$

$\omega = \frac{\pi}{30}$

And so our answer is $\omega = \frac{\pi}{30}$ radians/seconds.

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Example Question #5 : Angular Velocity

If you are walking around a circular track and your angular velocity through the angle $\theta = \frac{\pi}{6}$ is $\omega = \frac{\pi}{12}$ , how long would it take you to complete a full rotation?

Possible Answers:

22 seconds

24 seconds

12 seconds

2 seconds

Correct answer:

24 seconds

Explanation:

This is simply solving for an unknown. Here our unknown is time, . So we know that it takes you have an angular velocity of $\omega = \frac{\pi}{12}$ through the angle $\theta = \frac{\pi}{6}$ . Assuming that your angular velocity is consistent, we will be able to solve for . First we must manipulate our original angular velocity formula to solve for .

$\omega = \frac{\theta}{t}$

$t = \frac{\theta}{\omega}$

Now we will plug in our known value for $\omega$ and the angle of a full rotation. The angle of a full rotation is simply that of a full circle, $2\pi$ .

$t = \frac{\theta}{\omega}$

$t = \frac{2\pi}{\frac{\pi}{12}}$

$t = 2\pi (\frac{12}{\pi})$

t = 24

So it takes you 24 seconds to complete a full rotation

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Example Question #6 : Angular Velocity

You are on a carousel and have completed one full rotation in 15 seconds. What is your angular velocity?

Possible Answers:

$\omega = 15$ radians/seconds

$\omega = \frac{7\pi}{13}$ radians/seconds

$\omega = \frac{2\pi}{15}$ radians/seconds

$\omega = 2\pi$ radians/seconds

Correct answer:

$\omega = \frac{2\pi}{15}$ radians/seconds

Explanation:

Even though we are not directly given an angle measurement, we are told the time it takes to complete one full rotation. One rotation is the same as the angle of an entire circle, $2\pi$ . Now we have enough information to plug into the angular velocity formula to solve this problem.

$\omega = \frac{\theta}{t}$

$\omega = \frac{2\pi}{15}$

So the angular velocity is $\omega = \frac{2\pi}{15}$ radians/seconds

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Example Question #5 : Angle Applications

A wheel is making one full rotation with an angular velocity of $\frac{5\pi }{2}$ radians/seconds. How long will it take the wheel to make 10 full rotations?

Possible Answers:

8 seconds

1 second

24 seconds

20 seconds

Correct answer:

8 seconds

Explanation:

We know that the angular velocity to complete a full rotation (rotate through an angle of $2\pi$ ) is $\frac{5\pi}{2}$ radians/seconds. The angle of 10 full rotations will be 10 rotations through the angle $2\pi$ , so we multiply these two quantities.

$10 *2\pi = 20\pi$

And this will be our $\theta$ . Now we must manipulate our angular velocity formula to solve for .

$\omega = \frac{\theta}{t}$

$t = \frac{\theta}{\omega}$

Now we can plug in our $\theta$ for the 10 rotations and our angular velocity.

$t = \frac{\theta}{\omega}$

$t = \frac{20\pi}{\frac{5\pi}{2}}$

$t = 20\pi \frac{2}{5\pi}$

t = 8

And so it will take 8 seconds to make 10 rotations.

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Example Question #8 : Angular Velocity

If it takes you 3 seconds to rotate through an angle with an angular velocity of $\frac{8\pi}{9}$ , what is the measure of the angle you are rotating through?

Possible Answers:

$\theta = \frac{8\pi}{9}$

$3\pi$

$\theta = \frac{8\pi}{3}$

$\theta = \frac{\pi}{3}$

Correct answer:

$\theta = \frac{8\pi}{3}$

Explanation:

We are able to use the angular velocity formula and solve for the unknown $\theta$ .

$\omega = \frac{\theta}{t}$

$\omega t = \theta$

$\frac{8\pi}{9} (3) = \theta$ (making it so we are solving for $\theta$ )

$\frac{24\pi}{9} = \theta$

$\frac{8\pi}{3} = \theta$

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Example Question #1 : Area Of A Sector

Which of the following is the definition for a sector of a circle?

Possible Answers:

A section of a circle which is enclosed by a chord and the chord’s arc

Any section of a circle

A section of a circle which is enclosed by two radii joined at the center of the circle and the arc between the two radii

Any section of a circle that is triangular in shape

Correct answer:

A section of a circle which is enclosed by two radii joined at the center of the circle and the arc between the two radii

Explanation:

Below is an illustration of a sector of a circle. A sector is the area of a circle which has been enclosed by two radii and the arc between them. A sector is not to be confused with a segment of a circle. A segment is when the area enclosed by the chord of a circle and the arc of the chord.

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Example Question #2 : Area Of A Sector

Which of the following is the formula for the area of a sector?

Possible Answers:

$A = \theta \pi r^2$

$A = \frac{\theta\pi}{2} r^2$

$A = \frac{\theta}{2} r^2$

$A = 2\theta r^2$

Correct answer:

$A = \frac{\theta}{2} r^2$

Explanation:

When thinking about how to derive the formula for a sector, we must consider the angle of an entire circle. The angle of an entire circle, 360 degrees, is $2\pi$ and we know the area of a circle is $\pi r^2$ .

When considering a sector, this is only a portion of the entire circle, so it is a particular $\theta$ out of the entire $2\pi$ . We can plug this into our area for a circle and it will simplify to the area of a sector.

$\frac{\theta}{2\pi}\pi r^2$

$\frac{\theta}{2} r^2$

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Trigonometry : Angle Applications

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Angle Applications

Example Question #2 : Angle Applications

Example Question #3 : Angle Applications

Example Question #4 : Angle Applications

Example Question #5 : Angular Velocity

Example Question #6 : Angular Velocity

Example Question #5 : Angle Applications

Example Question #8 : Angular Velocity

Example Question #1 : Area Of A Sector

Example Question #2 : Area Of A Sector

All Trigonometry Resources

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