Question

How do you convert $-135$ degrees to radian?

Answer 1

Hint: Degree is a measurement of a plane angle in which one full rotation is ${360}^{\circ }$. A degree has the symbol ${}^{\circ }$. It is not only used for measuring angles but also for locating directions. Whereas, radian is the SI unit for measuring angles. Radian is the measurement of a plane angle subtended by a circular arc. It is denoted by $rad$ or ${}^C$.
Both degrees and radians are used to measure angles. One full revolution can be expressed by ${360}^{\circ }$ in degrees and $2\pi$ in radians. Hence, we have
$$2\pi ={360}^{\circ }$$
$\Rightarrow \pi ={\displaystyle \frac{{360}^{\circ }}{2}}$
$\Rightarrow \pi ={180}^{\circ }$
Dividing both sides by ${180}^{\circ }$, we get
$\Rightarrow {\displaystyle \frac{\pi }{{180}^{\circ }}}=1$
Now on using unitary method, we have
$Degrees\times {\displaystyle \frac{\pi }{{180}^{\circ }}}=Radians$.
Hence, Angle in $Radians$$=$ Angle in $Degrees\times {\displaystyle \frac{\pi }{{180}^{\circ }}}$
We will use the above-given formula to solve our question.

Complete step by step solution:
Given Angle in $Degrees=-{135}^{\circ }$. On substituting it in the formula, we will get
Angle in $Radians=-{135}^{\circ }\times {\displaystyle \frac{\pi }{{180}^{\circ }}}$
One arranging numerals together on the right-hand side of the equation, we will get
$\Rightarrow$ Angle in $Radians=-\pi \times {\displaystyle \frac{{135}^{\circ }}{{180}^{\circ }}}$
To simplify the fraction ${\displaystyle \frac{{135}^{\circ }}{{180}^{\circ }}}$, we have to find the GCD of $135$ and $180$.
So $GCD(135,180)=45$. So now we will divide both the numerator and the denominator of the fraction ${\displaystyle \frac{{135}^{\circ }}{{180}^{\circ }}}$ by ${45}^{\circ }$, such that
$\Rightarrow$ Angle in $Radians=-\pi \times {\displaystyle \frac{{\displaystyle \frac{{135}^{\circ }}{{45}^{\circ }}}}{{\displaystyle \frac{{180}^{\circ }}{{45}^{\circ }}}}}$
$\Rightarrow$ Angle in $Radians=-\pi \times {\displaystyle \frac{3}{4}}$
On further arranging, we will get,
$\Rightarrow$ Angle in $Radians=-{\displaystyle \frac{3\pi }{4}}$

Hence $-135$ degrees can be written in radians as $-{\displaystyle \frac{3\pi }{4}}$.

Note:
The Greatest Common Divisor (GCD) or the Greatest Common Factor, of two numbers, is the largest number that divides them both. The GCD of two numbers $a$ and $b$ is denoted as $GCD(a,b)$.