To find the measure of an angle that is coterminal with a [tex]\( 45^\circ \)[/tex] angle, we need to understand the concept of coterminal angles. Coterminal angles are angles that share the same initial and terminal sides after a full rotation, which is [tex]\( 360^\circ \)[/tex].
Given the four expressions:
- [tex]\( 45^\circ + 90^\circ \)[/tex]
- [tex]\( 45^\circ + 180^\circ \)[/tex]
- [tex]\( 45^\circ + 270^\circ \)[/tex]
- [tex]\( 45^\circ + 360^\circ \)[/tex]
We need to determine which of these will give us an angle that maintains the same position as the [tex]\( 45^\circ \)[/tex] angle after [tex]\( 360^\circ \)[/tex].
1. First Expression: [tex]\( 45^\circ + 90^\circ \)[/tex]
[tex]\[
45^\circ + 90^\circ = 135^\circ
\][/tex]
This angle is not coterminal with [tex]\( 45^\circ \)[/tex].
2. Second Expression: [tex]\( 45^\circ + 180^\circ \)[/tex]
[tex]\[
45^\circ + 180^\circ = 225^\circ
\][/tex]
This angle is also not coterminal with [tex]\( 45^\circ \)[/tex].
3. Third Expression: [tex]\( 45^\circ + 270^\circ \)[/tex]
[tex]\[
45^\circ + 270^\circ = 315^\circ
\][/tex]
This angle is not coterminal with [tex]\( 45^\circ \)[/tex].
4. Fourth Expression: [tex]\( 45^\circ + 360^\circ \)[/tex]
[tex]\[
45^\circ + 360^\circ = 405^\circ
\][/tex]
This angle is coterminal with [tex]\( 45^\circ \)[/tex] because it represents one full rotation plus the [tex]\( 45^\circ \)[/tex].
Therefore, the expression that finds the measure of an angle that is coterminal with a [tex]\( 45^\circ \)[/tex] angle is [tex]\( 45^\circ + 360^\circ \)[/tex]. This gives us the result:
[tex]\[
405^\circ
\][/tex]
