To find the remaining angle of a triangle when two angles are given, we need to recall that the sum of all interior angles in any triangle is always [tex]\(180^\circ\)[/tex]. Given the two known angles, we can follow these steps:
1. Convert the angle given in radians to degrees:
The given angle is [tex]\(\left(\frac{\pi}{6}\right)^{\circ}\)[/tex].
Knowing the conversion factor [tex]\(1 \text{ radian} = 180^\circ / \pi\)[/tex], we convert [tex]\(\frac{\pi}{6}\)[/tex] radians to degrees as follows:
[tex]\[
\left(\frac{\pi}{6}\right) \times \left(\frac{180^\circ}{\pi}\right) = 30^\circ
\][/tex]
So, [tex]\(\frac{\pi}{6}\)[/tex] radians is equivalent to [tex]\(30^\circ\)[/tex].
2. Identify the known angles in degrees:
The two given angles are:
[tex]\[
45^\circ \quad \text{and} \quad 30^\circ
\][/tex]
3. Calculate the remaining angle:
Using the fact that the sum of the angles in a triangle is [tex]\(180^\circ\)[/tex], we find the remaining angle by subtracting the sum of the known angles from [tex]\(180^\circ\)[/tex]:
[tex]\[
\text{Remaining angle} = 180^\circ - (45^\circ + 30^\circ)
\][/tex]
[tex]\[
\text{Remaining angle} = 180^\circ - 75^\circ = 105^\circ
\][/tex]
Thus, the remaining angle in the triangle is:
[tex]\[
105^\circ
\][/tex]
