Question

Find the degree measure of an angle of $3.54$ radians.

Answer 1

To convert the radian measure to degree measure, we first multiply the radian measure by ${\left( {\dfrac{{180}}{\pi }} \right)^ \circ }$so as to get the angle in degree measure. Then to convert degree to minute, we multiply the degrees by $60'$ to get the result in minutes and to convert the minutes to seconds by multiplying $60''$ to the given number which is in minutes the resultant number will be in second.

Complete step by step answer:
Radians and degrees are both units used for measuring angles. A circle is composed of $2\pi$ radians, which is the equivalent of $360$ degree. Both of these values represent going once around a circle. Therefore, $1\pi$ radian represents going around a semi-circle or covering ${180^ \circ }$.

This makes $\dfrac{{{{180}^ \circ }}}{\pi }$ the perfect conversion tool for converting radians to degrees.To convert from radians to degrees, you simply have to multiply the radian value by $\dfrac{{{{180}^ \circ }}}{\pi }$.Here in the question, we are required to convert $3.54$ radians. Firstly, we convert radian to degree by multiplying the radian measure by $3.54$.
$\Rightarrow 3.54 \times \dfrac{{{{180}^ \circ }}}{\pi }$
The value of $\pi$ is $\left( {\dfrac{{22}}{7}} \right)$.

Substituting in the value of $\pi$, we get,
$\Rightarrow 3.54 \times \dfrac{{{{180}^ \circ }}}{{\left( {\dfrac{{22}}{7}} \right)}}$
Simplifying the expression, we get,
$\Rightarrow 3.54 \times \dfrac{{{{180}^ \circ }}}{{22}} \times 7$
Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow 3.54 \times \dfrac{{{{90}^ \circ }}}{{11}} \times 7$
Simplifying the calculations and finding the degree measure,
$\Rightarrow 24.78 \times \dfrac{{{{90}^ \circ }}}{{11}}$
$\Rightarrow {\left( {\dfrac{{2230.2}}{{11}}} \right)^ \circ }$
$\Rightarrow {202.745^ \circ }$

So, the degree measure of the angle equivalent to $3.54$ radians is ${202.745^ \circ }$.

Note: One degree (°) is equal to $60$ minutes (') and one minute equals to sixty seconds. So, in turn, one degree is equal to $3600$ seconds. So, to convert decimal degrees to minutes, we may multiply $60$ to the decimal degree number and next to convert decimal minute to seconds we may multiply $60$ to the decimal minute. But since the question asked the measure of angle specifically in degrees, ${202.745^ \circ }$is the final answer.