Question

How do you convert $\;170$ degrees to radians?

Answer 1

Hint: Degrees and Radians are the units for the measurements of angle. The angle of a complete circle is measured in degrees as ${360^\circ }$ and in radians the angle is shown as $2\pi $ . As ${360^\circ }$ is equal to $2\pi $ , we can find the radians for ${170^\circ }$ with the help of cross multiplication.

Formula used:
Angle in radians = Angle in degrees $ \times \dfrac{\pi }{{180}}$

Complete step-by-step answer:
As we know, the angle subtended by a circle is considered ${360^\circ }$ .
Hence, one degree can be defined as the angle subtended by one three hundred and sixtieth part of the circumference of a circle
One radian can be defined as the angle subtended by an arc whose length is equal to the radius of the circle.
Radians can also be described as the length of an arc that subtends the angle divided by the radius of the arc.
Hence, if we consider a unit circle i.e. a circle with a radius of one unit,
Length of arc = Circumference of circle = $2\pi r$
As the radius of the circle is one unit,
Length of arc = $2\pi (1)$ = $2\pi $
Now, Radians = $\dfrac{{{\text{Length of arc}}}}{{{\text{Radius of arc}}}}$
Substituting length as $2\pi $ and radius as $\;1$
$ \Rightarrow {\text{Radians = }}\dfrac{{2\pi }}{1}$
$ \Rightarrow {\text{Radians = }}2\pi $
Now, we can say that angle subtended by a circle in degrees is ${360^\circ }$ and in radians is $2\pi $
$ \Rightarrow {360^\circ } = 2\pi $
Now, to find one degree is how much radians, divide both sides by $\;360$
$ \Rightarrow {1^\circ } = \dfrac{{2\pi }}{{360}}$
Eliminating the common factor from numerator and denominator
$ \Rightarrow {1^\circ } = \dfrac{\pi }{{180}}$
Now to find the value of ${170^\circ }$ in radians, multiply both sides with $\;170$
$ \Rightarrow {170^\circ } = \dfrac{\pi }{{180}} \times 170$
Eliminating the common factor, we can write the above equation as
$ \Rightarrow {170^\circ } = \dfrac{{17\pi }}{{18}}$

Hence, the value of ${170^\circ }$ in radians is $\dfrac{{17\pi }}{{18}}$.

Note:
As a short trick, we can remember for converting degrees to radians, we multiply the value by the factor $\dfrac{\pi }{{180}}$. Similarly, if we want to convert radians to degrees, we multiply the value by the factor $\dfrac{{180}}{\pi }$. Here, if further simplification is required, we can put the value of $\pi $ as $\;3.14$ .