To convert an angle from degrees to radians, we use the conversion factor [tex]\(\pi\)[/tex] radians [tex]\( = 180^\circ\)[/tex]. This means:
[tex]\[ 1^\circ = \frac{\pi}{180} \text{ radians} \][/tex]
Let's go through each conversion step by step.
### 1. [tex]\(40^\circ\)[/tex] to radians
First, we apply the conversion factor:
[tex]\[ 40^\circ \times \frac{\pi}{180} \][/tex]
Simplify the fraction:
[tex]\[ \frac{40\pi}{180} \][/tex]
Reduce the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 20:
[tex]\[ \frac{40\pi}{180} = \frac{2\pi}{9} \][/tex]
So, [tex]\(40^\circ\)[/tex] in radians is:
[tex]\[ 40^\circ = \frac{2\pi}{9} \][/tex]
### 2. [tex]\(-105^\circ\)[/tex] to radians
Similarly, apply the conversion factor:
[tex]\[ -105^\circ \times \frac{\pi}{180} \][/tex]
Simplify the fraction:
[tex]\[ \frac{-105\pi}{180} \][/tex]
Reduce the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 15:
[tex]\[ \frac{-105\pi}{180} = \frac{-7\pi}{12} \][/tex]
So, [tex]\(-105^\circ\)[/tex] in radians is:
[tex]\[ -105^\circ = \frac{-7\pi}{12} \][/tex]
### Final Answers
Thus, the measures converted to radians are:
1. [tex]\(40^\circ = \frac{2\pi}{9}\)[/tex]
2. [tex]\(-105^\circ = \frac{-7\pi}{12}\)[/tex]
