Question

Post two examples that are similar to questions to the following photo. Include the question and your solution. Example 1: Relates to converting between radians and degrees Example 2: Use the relationship a=rθ to solve for either the arc length (a), the radius (r) or the angle in radians (θ).

Answer 1

1) Given the angles below transform into degrees or radians as needed, round to two decimal points if necessary.

`\begin{gathered} 4\pi \\ 720\degree \\ 10\degree \\ (5)/(8)\pi \end{gathered}`

In general, 2pi=360°; then, 4pi radians refer to 2 revolutions around a circumference, as shown in the diagram below

Therefore, 4pi=0°=360°.

Similarly,

`\begin{gathered} (360\degree)/(2\pi)=(720\degree)/(x) \\ \Rightarrow x=2\pi((720)/(360))=2\pi *2=4\pi=0 \end{gathered}`

720°=0 radians

`\begin{gathered} (2\pi)/(360\degree)=(x)/(10\degree) \\ \Rightarrow x=2\pi((10)/(360))=(\pi)/(18) \end{gathered}`

10°=pi/18 radians.

`\begin{gathered} (x)/((5)/(8)\pi)=(360\degree)/(2\pi) \\ \Rightarrow x=360\degree(((5)/(8)\pi)/(2\pi))=360\degree((5)/(16))=112.5 \end{gathered}`

5pi/8=112.5°.

2) Given that the area of a circle is 30pi and an arc length on it is pi/2; find the central angle generated by such arc in radians, round to three decimal places if needed.

The area of a circle is given by

`\begin{gathered} A=\pi r^2 \\ r\rightarrow radius \end{gathered}`

Then, in our case,

`30\pi=r^2\pi\Rightarrow r=√(30)`

On the other hand, an arc length is given by the formula below

`arc=r\theta`

In our case,

`\begin{gathered} arc=(\pi)/(2) \\ \Rightarrow(\pi)/(2)=√(30)\theta \\ \Rightarrow\theta=(\pi)/(2√(30)) \\ \Rightarrow\theta\approx0.287\text{ radians} \end{gathered}`

The answer is 0.287 radians.