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 = \dfrac{{{{360}^{\circ}}}}{2}$
$ \Rightarrow \pi = {180^{\circ}}$
Dividing both sides by ${180^{\circ}}$, we get
$ \Rightarrow \dfrac{\pi }{{{{180}^{\circ}}}} = 1$
Now on using unitary method, we have
$Degrees \times \dfrac{\pi }{{{{180}^{\circ}}}} = Radians$.
Hence, Angle in $Radians$$ = $ Angle in $Degrees \times \dfrac{\pi }{{{{180}^{\circ}}}}$
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 $Radians = - {135^{\circ}} \times \dfrac{\pi }{{{{180}^{\circ}}}}$
One arranging numerals together on the right-hand side of the equation, we will get
$ \Rightarrow $ Angle in $Radians = - \pi \times \dfrac{{{{135}^{\circ}}}}{{{{180}^{\circ}}}}$
To simplify the fraction $\dfrac{{{{135}^{\circ}}}}{{{{180}^{\circ}}}}$, 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 $\dfrac{{{{135}^{\circ}}}}{{{{180}^{\circ}}}}$ by ${45^{\circ}}$, such that
$ \Rightarrow $ Angle in $Radians = - \pi \times \dfrac{{\dfrac{{{{135}^{\circ}}}}{{{{45}^{\circ}}}}}}{{\dfrac{{{{180}^{\circ}}}}{{{{45}^{\circ}}}}}}$
$ \Rightarrow $ Angle in $Radians = - \pi \times \dfrac{3}{4}$
On further arranging, we will get,
$ \Rightarrow $ Angle in $Radians = - \dfrac{{3\pi }}{4}$

Hence $ - 135$ degrees can be written in radians as $ - \dfrac{{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 $GCD(a,b)$.