Question

certain positive angles are marked in the given figure. find the corresponding radian or degrees measures.

$30^{\circ}=\square$ radians
(simplify your answer. type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)

$\frac{\pi}{4}$ radians $=\square^{\circ}$
(simplify your answer. do not include the degree symbol in your answer.)

$60^{\circ}=\square$ radians
(simplify your answer. type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)

$\frac{2\pi}{3}$ radians $=\square^{\circ}$
(simplify your answer. do not include the degree symbol in your answer.)

$\frac{3\pi}{4}$ radians $=\square^{\circ}$
(simplify your answer. do not include the degree symbol in your answer.)

$150^{\circ}=\square$ radians
(simplify your answer. type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)

$180^{\circ}=\square$ radians
(simplify your answer. type an exact answer, using $\pi$ as needed. use integers or fractions for any numbers in the expression.)

$210^{\circ}=\square$ radians

Explanation:

Step1: Convert \(30^\circ\) to radians

We know the conversion formula: \(1^\circ=\frac{\pi}{180}\) radians. So for \(30^\circ\), we multiply \(30\) by \(\frac{\pi}{180}\).
\(30\times\frac{\pi}{180}=\frac{\pi}{6}\)

Step2: Convert \(\frac{\pi}{4}\) radians to degrees

Using the formula \(1\) radian \(=\frac{180^\circ}{\pi}\), so \(\frac{\pi}{4}\) radians is \(\frac{\pi}{4}\times\frac{180^\circ}{\pi}\). The \(\pi\) cancels out, and \(\frac{180}{4} = 45\).

Step3: Convert \(60^\circ\) to radians

Using \(1^\circ=\frac{\pi}{180}\) radians, multiply \(60\) by \(\frac{\pi}{180}\).
\(60\times\frac{\pi}{180}=\frac{\pi}{3}\)

Step4: Convert \(\frac{2\pi}{3}\) radians to degrees

Using \(1\) radian \(=\frac{180^\circ}{\pi}\), so \(\frac{2\pi}{3}\times\frac{180^\circ}{\pi}\). The \(\pi\) cancels out, and \(\frac{2\times180}{3}=120\).

Step5: Convert \(\frac{3\pi}{4}\) radians to degrees

Using \(1\) radian \(=\frac{180^\circ}{\pi}\), so \(\frac{3\pi}{4}\times\frac{180^\circ}{\pi}\). The \(\pi\) cancels out, and \(\frac{3\times180}{4} = 135\).

Step6: Convert \(150^\circ\) to radians

Using \(1^\circ=\frac{\pi}{180}\) radians, multiply \(150\) by \(\frac{\pi}{180}\).
\(150\times\frac{\pi}{180}=\frac{5\pi}{6}\)

Step7: Convert \(180^\circ\) to radians

Using \(1^\circ=\frac{\pi}{180}\) radians, multiply \(180\) by \(\frac{\pi}{180}\), which gives \(\pi\).

Step8: Convert \(210^\circ\) to radians

Using \(1^\circ=\frac{\pi}{180}\) radians, multiply \(210\) by \(\frac{\pi}{180}\).
\(210\times\frac{\pi}{180}=\frac{7\pi}{6}\)

Answer:

\(30^\circ=\boldsymbol{\frac{\pi}{6}}\) radians; \(\frac{\pi}{4}\) radians \(=\boldsymbol{45}\)°; \(60^\circ=\boldsymbol{\frac{\pi}{3}}\) radians; \(\frac{2\pi}{3}\) radians \(=\boldsymbol{120}\)°; \(\frac{3\pi}{4}\) radians \(=\boldsymbol{135}\)°; \(150^\circ=\boldsymbol{\frac{5\pi}{6}}\) radians; \(180^\circ=\boldsymbol{\pi}\) radians; \(210^\circ=\boldsymbol{\frac{7\pi}{6}}\) radians