Angles in the Coordinate Plane - Pre-Calculus

Card 0 of 44

Question

Convert $\frac{8\pi}{3}$ radians to degrees.

Answer

Write the conversion factor between radians and degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

Cancel the radians unit by using dimensional analysis.

$(\frac{8\pi}{3} \textup{ radians })(\frac{180 \textup{ degrees}}{\pi \textup{ radians}}) = 480 \textup{ degrees}$

Compare your answer with the correct one above

Question

Convert $\frac{3}{50}\pi$ to degrees.

Answer

Write the conversion factor of radians and degrees.

$\pi \textup{ radians}= 180 \textup{ degrees}$

Substitute the degree measure into $\pi$ .

$\frac{3}{50}\pi= \frac{3}{50}(180)= 10.8$

Compare your answer with the correct one above

Question

Determine the angle $\theta$ in degres made in the plane by connecting a line segment from the origin to .

Assume $\theta >0$

Answer

Firstly, since the point is in the 3rd quadrant, it'll be between and . If we take to be the horizontal, we can form a triangle with base and leg of values and . Solving for the angle in the 3rd quadrant given by $\gamma$ ,

$tan(\gamma)=\frac{-1}{-1}=1$

$\gamma=45^o$

Since this angle is made by assuming to be the horizontal, the total angle measure $\theta$ is going to be:

$\theta=180^o+45^o=225^o$

Compare your answer with the correct one above

Question

Find the degree measure of $\frac{11}{8} \pi$ radians. Round to the nearest integer.

Answer

In order to solve for the degree measure from radians, replace the $\pi$ radians with 180 degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

$\frac{11}{8} (180) = \frac{11\times 45}{2} = \frac{495}{2}= 247.5$

The nearest degree is .

Compare your answer with the correct one above

Question

Given a triangle, the first angle is three times the value of the second angle. The third angle is $\frac{\pi}{3} \textup{ radians}$ . What is the value of the second largest angle in degrees?

Answer

A triangle has three angles that will add up to degrees.

Convert the radians angle to degrees by substituting $180 \textup{ degrees}$ for every $\pi \textup{ radian}$ .

$\frac{180}{3} = 60 \textup{ degrees}$

The third angle is 60 degrees.

Let the second angle be . The first angle three times the value of the second angle is . Set up an equation that sums the three angles to $180 \textup{ degrees}$ .

Solve for .

Substitute for the first angle and second angle.

The second angle is:

The first angle is:

The three angles are:

The second highest angle is:

Compare your answer with the correct one above

Question

Find the coterminal angle of 15 degrees.

Answer

The coterminal angles can be positive or negative. To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

All of these angles are coterminal angles.

Compare your answer with the correct one above

Question

Of the given answers, what of the following is a coterminal angle of $\pi$ radians?

Answer

To find the coterminal angle of an angle, simply add or subtract $2\pi$ radians, or 360 degrees as many times as needed.

$\pi+2\pi=3\pi$

$\pi-2\pi=-\pi$

$3\pi+2\pi=5\pi$

$-\pi-2\pi=-3\pi$

These are all coterminal angles to $\pi$ radians.

Out of the given answers, $3\pi$ is the only possible answer.

Compare your answer with the correct one above

Question

Find the coterminal angle of $10^\circ$ , if possible.

Answer

In order to find a coterminal angle, or angles of the given angle, simply add or subtract 360 degrees of the terminal angle as many times as possible.

The only correct answer is .

Compare your answer with the correct one above

Question

Of the following choices, find a coterminal angle of $\pi$ .

Answer

In order to find a coterminal angle, simply add or subtract $2\pi$ radians to the given angle as many times as possible.

$\theta=\pi+2\pi k$

The possible angles after adding increments of $2\pi$ radians are:

$\pi, 3\pi,5\pi,7\pi...$

$\theta=\pi-2\pi k$

The possible angles after subtracting decrements of $2\pi$ radians are:

$\pi,-\pi,-3\pi,-5\pi...$

Out of the given possibilities, only $-5\pi$ is a valid answer.

Compare your answer with the correct one above

Question

Find the coterminal angle of 15 degrees in standard position from the following answers.

Answer

To determine the coterminal angle, simply add or subtract increments or decrements of 360 degrees to the given angle.

For :

These angles can all be coterminal to 15 degrees. The only answer is .

Compare your answer with the correct one above

Question

Which of the following angles is coterminal to $65^{\circ}$ ?

Answer

Which of the following angles is coterminal to $65^{\circ}$ ?

Coterminal angles are angles which start and end at the same point. In other words, they share both their starting and ending point. Note, this doesn't require them to be the same angle.

For instance, $120^{\circ}$ is coterminal with $-240^{\circ}$ , because they both start on the positive x-axis, and end at the same place in quadrant 2.

So, we want to find an angle that ends at the same place in quadrant 1 as $65^{\circ}$ . Of the answer choices, only 1 ends in quadrant 1, so that one must be our answer: $-295^{\circ}$

Compare your answer with the correct one above

Question

Convert $\frac{8\pi}{3}$ radians to degrees.

Answer

Write the conversion factor between radians and degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

Cancel the radians unit by using dimensional analysis.

$(\frac{8\pi}{3} \textup{ radians })(\frac{180 \textup{ degrees}}{\pi \textup{ radians}}) = 480 \textup{ degrees}$

Compare your answer with the correct one above

Question

Convert $\frac{3}{50}\pi$ to degrees.

Answer

Write the conversion factor of radians and degrees.

$\pi \textup{ radians}= 180 \textup{ degrees}$

Substitute the degree measure into $\pi$ .

$\frac{3}{50}\pi= \frac{3}{50}(180)= 10.8$

Compare your answer with the correct one above

Question

Determine the angle $\theta$ in degres made in the plane by connecting a line segment from the origin to .

Assume $\theta >0$

Answer

Firstly, since the point is in the 3rd quadrant, it'll be between and . If we take to be the horizontal, we can form a triangle with base and leg of values and . Solving for the angle in the 3rd quadrant given by $\gamma$ ,

$tan(\gamma)=\frac{-1}{-1}=1$

$\gamma=45^o$

Since this angle is made by assuming to be the horizontal, the total angle measure $\theta$ is going to be:

$\theta=180^o+45^o=225^o$

Compare your answer with the correct one above

Question

Find the degree measure of $\frac{11}{8} \pi$ radians. Round to the nearest integer.

Answer

In order to solve for the degree measure from radians, replace the $\pi$ radians with 180 degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

$\frac{11}{8} (180) = \frac{11\times 45}{2} = \frac{495}{2}= 247.5$

The nearest degree is .

Compare your answer with the correct one above

Question

Given a triangle, the first angle is three times the value of the second angle. The third angle is $\frac{\pi}{3} \textup{ radians}$ . What is the value of the second largest angle in degrees?

Answer

A triangle has three angles that will add up to degrees.

Convert the radians angle to degrees by substituting $180 \textup{ degrees}$ for every $\pi \textup{ radian}$ .

$\frac{180}{3} = 60 \textup{ degrees}$

The third angle is 60 degrees.

Let the second angle be . The first angle three times the value of the second angle is . Set up an equation that sums the three angles to $180 \textup{ degrees}$ .

Solve for .

Substitute for the first angle and second angle.

The second angle is:

The first angle is:

The three angles are:

The second highest angle is:

Compare your answer with the correct one above

Question

Find the coterminal angle of 15 degrees.

Answer

The coterminal angles can be positive or negative. To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

All of these angles are coterminal angles.

Compare your answer with the correct one above

Question

Of the given answers, what of the following is a coterminal angle of $\pi$ radians?

Answer

To find the coterminal angle of an angle, simply add or subtract $2\pi$ radians, or 360 degrees as many times as needed.

$\pi+2\pi=3\pi$

$\pi-2\pi=-\pi$

$3\pi+2\pi=5\pi$

$-\pi-2\pi=-3\pi$

These are all coterminal angles to $\pi$ radians.

Out of the given answers, $3\pi$ is the only possible answer.

Compare your answer with the correct one above

Question

Find the coterminal angle of $10^\circ$ , if possible.

Answer

In order to find a coterminal angle, or angles of the given angle, simply add or subtract 360 degrees of the terminal angle as many times as possible.

The only correct answer is .

Compare your answer with the correct one above

Question

Of the following choices, find a coterminal angle of $\pi$ .

Answer

In order to find a coterminal angle, simply add or subtract $2\pi$ radians to the given angle as many times as possible.

$\theta=\pi+2\pi k$

The possible angles after adding increments of $2\pi$ radians are:

$\pi, 3\pi,5\pi,7\pi...$

$\theta=\pi-2\pi k$

The possible angles after subtracting decrements of $2\pi$ radians are:

$\pi,-\pi,-3\pi,-5\pi...$

Out of the given possibilities, only $-5\pi$ is a valid answer.

Compare your answer with the correct one above

Tap the card to reveal the answer