Understanding Radians and Conversions

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Math › Understanding Radians and Conversions

Questions 1 - 10

1

If Angle equals $72^{\circ}$ , what is the equivalent angle in radians (to the nearest hundredth)?

$1.26\ radians$

$4125.29\ radians$

$4.125\ radians$

$1256.6\ radians$

$0.628\ radians$

Explanation

To convert between radians and degrees, it is important to remember that:

$\pi, rad = 180^{\circ}$

With this relationship in mind, we can convert from degrees to radians with the following formula:

$72^{\circ}\cdot \frac{\pi}{180^{\circ}} = 1.26,rad$

2

How many radians are in $270^\circ$ ?

$\frac{3\pi}{2}$

$2\pi$

$\pi$

$\frac{2\pi}{3}$

$\frac{5\pi}{2}$

Explanation

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{270^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$270^\circ*\pi=180^\circ*x$

Isolate :

$\frac{270^\circ}{180^\circ}\pi=x$

$\frac{3\pi}{2}=x$

3

If angle equals $136^{\circ}$ , what is the equivalent angle in radians (to the nearest hundredth)?

$2.37\ rad$

$23.7\ rad$

$1.18\ rad$

$7.79\ rad$

$136\ rad$

Explanation

To convert between radians and degrees, it is important to remember that:

$\pi, rad = 180^{\circ}$

With this relationship in mind, we can convert from degrees to radians with the following formula:

$136^{\circ}\cdot \frac{\pi}{180^{\circ}} = 2.37,rad$

4

A point has Cartesian coordinates . Rewrite this as an ordered pair in the polar coordinate plane, rounding the coordinates to the nearest hundredth.

Explanation

Set . Calculate the polar coordinates $(r,\theta )$ as follows:

$r = \sqrt{x^{2}+y^{2}}= \sqrt{1^{2}+3^{2}}=\sqrt{10} \approx 3.16$

$\theta = \arctan \frac{y}{x}= \arctan \frac{3}{1} = \arctan 3 \approx 1.25$

5

How many degrees are in $\frac{\pi}{3}$ radians?

$60^\circ$

$30^\circ$

$540^\circ$

$45^\circ$

$15^\circ$

Explanation

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{\pi}{3}$

Divide both sides by $\pi$ :

$x^\circ=\frac{180^\circ}{3}$

$x^\circ=60^\circ$

6

What is the value of the angle $\frac{3\pi}{2}$ radians when converted to degrees?

Explanation

The answer can be found using the conversion of 1 radian equals $\frac{180}{\pi}$ degrees. Multiplying $\frac{3\pi}{2}$ by this conversion factor gives 270 degrees.

7

If $\angle A$ equals $45^{\circ}$ , what is the equivalent angle in radians (to the nearest hundredth)?

$0.79\ rad$

$0.56\ rad$

$3.14\ rad$

$1.57\ rad$

$4.71\ rad$

Explanation

To convert between radians and degrees, it is important to remember that:

$\pi, rad = 180^{\circ}$

With this relationship in mind, we can convert from degrees to radians with the following formula:

$45^{\circ}\cdot \frac{\pi}{180^{\circ}} = 0.79,rad$

8

How many degrees are in $\frac{8\pi}{3}$ radians?

$480^\circ$

$460^\circ$

$420^\circ$

$30^\circ$

$120^\circ$

Explanation

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{8\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{8\pi}{3}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{8}{3}$

$x^\circ=480^\circ$

9

If $\angle B$ is $275^{\circ}$ , what is the equivalent angle in radians (to the nearest hundredth)?

$4.80\ rad$

$2.40\ rad$

$1.37\ rad$

$9.60\ rad$

$3.27\ rad$

Explanation

To convert between radians and degrees, it is important to remember that:

$\pi, rad = 180^{\circ}$

With this relationship in mind, we can convert from degrees to radians with the following formula:

$275^{\circ}\cdot \frac{\pi}{180^{\circ}} = 4.80,rad$

10

How many degrees are in $\frac{3\pi}{4}$ radians?

$135^\circ$

$45^\circ$

$180^\circ$

$225^\circ$

$405^\circ$

Explanation

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{3\pi}{4}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{3\pi}{4}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{3}{4}$

$x^\circ=135^\circ$

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