Question

Express the following angles in degrees.
1) \[{\left( {\dfrac{{5\pi }}{{12}}} \right)^c}\]
2) \[ - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c}\]
3) \[{\left( {\dfrac{\pi }{3}} \right)^c}\]
4) \[\dfrac{{5\pi }}{6}\]
5) \[\dfrac{{2\pi }}{9}\]
6) \[\dfrac{{7\pi }}{{24}}\]

Answer 1

Hint:
Here we will find the given angle in degree by using the relation between angle and degree. We will convert the radian into a degree by multiplying the converting ratio to the measure of angles. We will simplify it further to get the required answer.

Complete step by step solution:
1) \[{\left( {\dfrac{{5\pi }}{{12}}} \right)^c}\]
We know that \[1{\rm{rad}} = {\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\].
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[{\left( {\dfrac{{5\pi }}{{12}}} \right)^c} = {\left( {\dfrac{{5\pi }}{{12}} \times \dfrac{{180}}{\pi }} \right)^ \circ }\]
Simplifying the expression, we get
\[ \Rightarrow {\left( {\dfrac{{5\pi }}{{12}}} \right)^c} = {\left( {5 \times 15} \right)^ \circ }\]
Multiplying the terms, we get
\[ \Rightarrow {\left( {\dfrac{{5\pi }}{{12}}} \right)^c} = {75^ \circ }\]
So angle \[{\left( {\dfrac{{5\pi }}{{12}}} \right)^c}\] is equal to \[{75^ \circ }\]

2) \[ - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c}\]
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[ - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c} - {\left( {\dfrac{{7\pi }}{{12}} \times \dfrac{{180}}{\pi }} \right)^ \circ }\]
Simplifying the expression, we get
\[ \Rightarrow - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c} = {\left( { - 7 \times 15} \right)^ \circ }\]
Multiplying the terms, we get
\[ \Rightarrow - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c} = - {105^ \circ }\]
As the above angle is negative so we will add \[{360^ \circ }\] to it to get the value as,
\[\begin{array}{l} \Rightarrow - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c} = {360^ \circ } - {105^ \circ }\\ \Rightarrow - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c} = {255^ \circ }\end{array}\]
So, angle \[ - {\left( {\dfrac{{7\pi }}{{12}}} \right)^c}\] is equal to \[{255^ \circ }\].

3) \[{\left( {\dfrac{\pi }{3}} \right)^c}\]
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[\begin{array}{l}{\left( {\dfrac{\pi }{3}} \right)^c} = {\left( {\dfrac{\pi }{3} \times \dfrac{{180}}{\pi }} \right)^ \circ }\\ \Rightarrow {\left( {\dfrac{\pi }{3}} \right)^c} = {60^ \circ }\end{array}\]
So, angle \[{\left( {\dfrac{\pi }{3}} \right)^c}\] is equal to \[{60^ \circ }\].

4) \[\dfrac{{5\pi }}{6}\]
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[\begin{array}{l}\dfrac{{5\pi }}{6} = {\left( {\dfrac{{5\pi }}{6} \times \dfrac{{180}}{\pi }} \right)^ \circ }\\ \Rightarrow \dfrac{{5\pi }}{6} = {\left( {5 \times 30} \right)^ \circ }\end{array}\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{{5\pi }}{6} = {150^ \circ }\]
So, angle\[\dfrac{{5\pi }}{6}\] is equal to \[{150^ \circ }\].

5) \[\dfrac{{2\pi }}{9}\]
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[\begin{array}{l}\dfrac{{2\pi }}{9} = {\left( {\dfrac{{2\pi }}{9} \times \dfrac{{180}}{\pi }} \right)^ \circ }\\ \Rightarrow \dfrac{{2\pi }}{9} = {\left( {2 \times 20} \right)^ \circ }\end{array}\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{{2\pi }}{9} = {40^ \circ }\]
So, angle \[\dfrac{{2\pi }}{9}\] is equal to \[{40^ \circ }\].

6) \[\dfrac{{7\pi }}{{24}}\]
We will multiply the above angle by \[{\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]. Therefore, we get
\[\begin{array}{l}\dfrac{{7\pi }}{{24}} = {\left( {\dfrac{{7\pi }}{{24}} \times \dfrac{{180}}{\pi }} \right)^ \circ }\\ \Rightarrow \dfrac{{7\pi }}{{24}} = {\left( {7 \times 7.5} \right)^ \circ }\end{array}\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{{7\pi }}{{24}} = {52.5^ \circ }\]
So, angle \[\dfrac{{7\pi }}{{24}}\] is equal to \[{52.5^ \circ }\].

Note:
Radian is used as an S.I unit to measure angles. It describes a plane angle made by a circular arc which is the length of the arc divided by the radius of the arc. A degree is used to measure a plane angle in which one full rotation is equal to 360 degrees. Degree is not an S.I unit but we can see it in SI brochure as an accepted unit.