To determine which of the given angles are coterminal with [tex]\( 145^\circ \)[/tex], we need to find angles that differ from [tex]\( 145^\circ \)[/tex] by a multiple of [tex]\( 360^\circ \)[/tex], since [tex]\( 360^\circ \)[/tex] represents a full rotation. This means for an angle [tex]\( \theta \)[/tex] to be coterminal with [tex]\( 145^\circ \)[/tex], the difference [tex]\( \theta - 145 \)[/tex] should be an integer multiple of [tex]\( 360 \)[/tex].
Let's check each given angle:
1. For [tex]\( -575^\circ \)[/tex]:
[tex]\[
-575 - 145 = -720
\][/tex]
Since [tex]\(-720\)[/tex] is a multiple of [tex]\( 360 \)[/tex] ([tex]\( -720 = 360 \times -2 \)[/tex]), [tex]\( -575^\circ \)[/tex] is coterminal with [tex]\( 145^\circ \)[/tex].
2. For [tex]\( -215^\circ \)[/tex]:
[tex]\[
-215 - 145 = -360
\][/tex]
Since [tex]\(-360\)[/tex] is a multiple of [tex]\( 360 \)[/tex] ([tex]\( -360 = 360 \times -1 \)[/tex]), [tex]\( -215^\circ \)[/tex] is coterminal with [tex]\( 145^\circ \)[/tex].
3. For [tex]\( -145^\circ \)[/tex]:
[tex]\[
-145 - 145 = -290
\][/tex]
[tex]\(-290\)[/tex] is not a multiple of [tex]\( 360 \)[/tex], so [tex]\( -145^\circ \)[/tex] is not coterminal with [tex]\( 145^\circ \)[/tex].
4. For [tex]\( -35^\circ \)[/tex]:
[tex]\[
-35 - 145 = -180
\][/tex]
[tex]\(-180\)[/tex] is not a multiple of [tex]\( 360 \)[/tex], so [tex]\( -35^\circ \)[/tex] is not coterminal with [tex]\( 145^\circ \)[/tex].
5. For [tex]\( 215^\circ \)[/tex]:
[tex]\[
215 - 145 = 70
\][/tex]
[tex]\( 70 \)[/tex] is not a multiple of [tex]\( 360 \)[/tex], so [tex]\( 215^\circ \)[/tex] is not coterminal with [tex]\( 145^\circ \)[/tex].
6. For [tex]\( 415^\circ \)[/tex]:
[tex]\[
415 - 145 = 270
\][/tex]
[tex]\( 270 \)[/tex] is not a multiple of [tex]\( 360 \)[/tex], so [tex]\( 415^\circ \)[/tex] is not coterminal with [tex]\( 145^\circ \)[/tex].
7. For [tex]\( 505^\circ \)[/tex]:
[tex]\[
505 - 145 = 360
\][/tex]
Since [tex]\( 360 \)[/tex] is a multiple of [tex]\( 360 \)[/tex] ([tex]\( 360 = 360 \times 1 \)[/tex]), [tex]\( 505^\circ \)[/tex] is coterminal with [tex]\( 145^\circ \)[/tex].
8. For [tex]\( 865^\circ \)[/tex]:
[tex]\[
865 - 145 = 720
\][/tex]
Since [tex]\( 720 \)[/tex] is a multiple of [tex]\( 360 \)[/tex] ([tex]\( 720 = 360 \times 2 \)[/tex]), [tex]\( 865^\circ \)[/tex] is coterminal with [tex]\( 145^\circ \)[/tex].
Thus, the angles that are coterminal with [tex]\( 145^\circ \)[/tex] are:
[tex]\[
-575^\circ, -215^\circ, 505^\circ, 865^\circ
\][/tex]
