Question

How to convert $285$ degrees to radians $?$

Answer 1

Hint: We convert degrees to the radians. First we multiply by $\dfrac{\pi }{{{{180}^ \circ }}}$ . And then take a common factor and cancel it. Finally we get a radians. In a unit circle, a full rotation corresponds to ${360^ \circ }$ whereas, a full rotation is related to $2\pi $ radians, the circumference of the unit circle. Thus, we have the following relations.
$2\pi $ radians is ${360^ \circ }$ .
 $1$ degree$ = \dfrac{\pi }{{180}}$ radians.
We just use multiplication and division.

Complete step-by-step solution:
To convert degrees to radians.
The given degrees is $285$
Multiply by $\dfrac{\pi }{{{{180}^ \circ }}}$ , hence we get
Since a full circle is ${360^ \circ }$ or $2\pi $ radians.
 ${285^ \circ } \times \dfrac{\pi }{{{{180}^ \circ }}}$
Factor $15$ out of $285$ , hence we get
 $15(19) \times \dfrac{\pi }{{180}}$radians
Factor $15$ out of $180$, hence we get
 $15 \times 19 \times \dfrac{\pi }{{15 \times 12}}$
Cancel the common factor, hence we get
 $1\not{5} \times 19 \times \dfrac{\pi }{{1\not{5} \times 12}}$
Rewrite the expression, we get
 $19 \times \dfrac{\pi }{{12}}$ radians
Combine $19$ and $\dfrac{\pi }{{12}}$, hence we get
 $\dfrac{{19\pi }}{{12}}$ radians.

Therefore 285 degrees is equal to $\dfrac{{19\pi }}{{12}}$ radians.

Note: Relationship between degree and radians measure, we have degree and radians units to measure angels.
One measuring unit is better than another if it can be defined more simply and intuitively. For example, in measuring temperature, the Celsius unit is better than Fahrenheit as Celsius was defined using ${0^ \circ }$ and ${100^ \circ }$ for freezing and boiling points of water. radians measure is better for conversation and calculations.
Greek mathematicians observed the relation of $\pi $ which arises from the circumference of a circle and thus, $\pi $plays a crucial role in radians measure.
In a unit circle, a full rotation corresponds to ${360^ \circ }$ whereas, a full rotation is related to $2\pi$ radians, the circumference of the unit circle. Thus, we have the following relations.
 $2\pi $ radians is ${360^ \circ }$ .