RELATION BETWEEN DEGREES AND RADIANS

Relation between Degrees and Radians

A radian is a measure of an angle, indicating the ratio between the arc length and the radius of a circle, which is expressed as follows:

{\rm{Radians}} = \frac{{{\rm{arc length }}}}{{{\rm{radius }}}}

Since, the arc length of a full circle is the same as the circumference of a circle, which is written as, then, the radian of the full circle \left( {360^\circ } \right) is:

{\rm{Radians}} = \frac{{{\rm{arc length }}}}{{{\rm{radius }}}} = \frac{{2\pi r}}{r} = 2\pi

 

The radian measure of angle:

The circular measure of angle means the number of radians a circle contains. Hence, the radian (circular) measure of a right angle is\frac{\pi }{2}.

clip_image012

The formula for converting radians to degrees:

{\rm{degrees}} = \frac{{radians \times 180}}{\pi }

Example

Convert 2.4 radians measure into degree measure

Use the formula for converting radians to degrees:

{\rm{degrees}} = \frac{{radians \times 180}}{\pi }

Substitute the radian as follows:

\begin{array}{c}
{\rm{degrees}} = \frac{{2.4 \times 180}}{\pi }\\
 = \frac{{423^\circ }}{\pi }\\
 = 134.65^\circ 
\end{array}

The formula for converting degrees to radians:

radians = \frac{{{\rm{degrees}} \times \pi }}{{180}}

Example

Convert 160^\circ into radian measure

Use the formula for converting degrees to radians:

radians = \frac{{{\rm{degrees}} \times \pi }}{{180}}

Substitute the degree as follows:

\begin{array}{c}
radians = \frac{{160 \times \pi }}{{180}}\\
 = \frac{{8 \times \pi }}{9}\\
 = \frac{{25.13}}{9}\\
 = 2.792\,{\rm{radians}}
\end{array}

The relationship between degree measure and radian measures for some standard angles are given below:

Degrees

Radians

clip_image027

clip_image029

clip_image031

clip_image033

clip_image035

clip_image037

clip_image039

clip_image041

clip_image043

clip_image010[1]

clip_image046

clip_image048

clip_image050

clip_image052

clip_image054

clip_image056

clip_image058

clip_image060

clip_image062

clip_image064

clip_image066

clip_image068

Question 1:

Convert 1 radian into degree measure.

Solution

\begin{array}{l}
{1^c}\, = \,{\left( {1 \times \frac{{180}}{\pi }} \right)^{\rm{o}}}\\
\,\,\,\,\, = \,{\left( {\frac{{180\, \times 7}}{{22}}} \right)^{\rm{o}}}\\
\,\,\,\,\, = \,{\left( {\frac{{630}}{{11}}} \right)^{\rm{o}}}\\
\,\,\,\,\, = \,{\left( {57\frac{3}{{11}}} \right)^{\rm{o}}}\\
\,\,\,\,\, = \,{57^{\rm{o}}}{\left( {\frac{3}{{11}}\, \times \,60} \right)^'}\\
\,\,\,\,\, = {57^{\rm{o}}}{\left( {\frac{{180}}{{11}}} \right)^'}\,\\
\,\,\,\,\, = \,{57^{\rm{o}}}16'{\left( {\frac{4}{{11}}\, \times \,60} \right)^'}\\
\,\,\,\,\, = {57^{\rm{o}}}16'{\left( {\frac{{240}}{{11}}} \right)^'}\\
\,\,\,\, = {57^{\rm{o}}}16'21.8''
\end{array}

Question 2:

Convert 5. 2 radians measure into degree measure

Solution

Use the formula for converting radians to degrees:

{\rm{degrees}} = \frac{{radians \times 180}}{\pi }

Substitute the radian as follows:

\begin{array}{c}
{\rm{degrees}} = \frac{{5.2 \times 180}}{\pi }\\
 = \frac{{936^\circ }}{\pi }\\
 = \left( {\frac{{936^\circ \, \times \,7}}{{22}}} \right)\\
 = {\left( {\frac{{3276}}{{11}}} \right)^{\rm{o}}}\\
 = {\left( {297\frac{9}{{11}}} \right)^{\rm{o}}}\\
 = {297^{\rm{o}}}{\left( {\frac{9}{{11}}\, \times \,60} \right)^'}\\
 = {297^{\rm{o}}}\,{\left( {\frac{{540}}{{11}}} \right)^'}\\
 = {297^{\rm{o}}}{\left( {49\frac{1}{{11}}} \right)^'}\\
 = {297^{\rm{o}}}49'\,{\left( {\frac{1}{{11}}\, \times 60} \right)^{''}}\\
 = {297^{\rm{o}}}49'{\left( {\frac{{60}}{{11}}} \right)^{''}}\\
 = {297^{\rm{o}}}49'5.45''
\end{array}

Therefore, 5. 2 radian is converted to .

Question 3:

Convert200^\circ into radian measure

Solution

Use the formula for converting degrees to radians:

radians = \frac{{{\rm{degrees}} \times \pi }}{{180}}

Substitute the degree as follows:

\begin{array}{c}
radians = \frac{{200 \times \pi }}{{180}}\\
 = \frac{{10 \times \pi }}{9}\\
 = \frac{{31.42}}{9}\\
 = {\rm{radians}}
\end{array}

Therefore, 200^\circ is converted to 3.49 radians.

Post Author: E-Maths