How many radians is [tex]\(-135^{\circ}\)[/tex]?

A. [tex]\(-\frac{1}{4} \pi\)[/tex]
B. [tex]\(\frac{3}{4} \pi\)[/tex]
C. [tex]\(-\frac{3}{4} \pi\)[/tex]
D. [tex]\(-\frac{4}{3} \pi\)[/tex]



Answer :

To convert an angle from degrees to radians, we use the relationship:

[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]

Given the angle is [tex]\( -135^\circ \)[/tex], we can express this conversion with the formula:

[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]

Substituting [tex]\(-135^\circ\)[/tex] into the formula:

[tex]\[ \text{radians} = -135 \times \frac{\pi}{180} \][/tex]

To simplify this:

[tex]\[ \text{radians} = -135 \times \frac{\pi}{180} = -135 \times \frac{\pi}{180} \][/tex]
[tex]\[ = - \frac{135}{180} \pi \][/tex]

We can simplify the fraction [tex]\(\frac{135}{180}\)[/tex]:

[tex]\[ \frac{135}{180} = \frac{135 \div 45}{180 \div 45} = \frac{3}{4} \][/tex]

So:

[tex]\[ \text{radians} = -\frac{3}{4} \pi \][/tex]

Hence, the radian measure corresponding to [tex]\( -135^\circ \)[/tex] is:

[tex]\[ -\frac{3}{4} \pi \][/tex]

Therefore, the correct answer is:

[tex]\[ -\frac{3}{4} \pi \][/tex]