To determine the measure of an angle that is coterminal with a given angle, we need to find the measure that, when wrapped around the circle, ends up in the range of 0° to 360°.
Given the original angle of [tex]\( 300^{\circ} \)[/tex], we are provided with four different expressions to evaluate:
1. [tex]\( 300^{\circ} - 860^{\circ} \)[/tex]
2. [tex]\( 300^{\circ} - 840^{\circ} \)[/tex]
3. [tex]\( 300^{\circ} - 740^{\circ} \)[/tex]
4. [tex]\( 300^{\circ} - 720^{\circ} \)[/tex]
Let’s find the resulting angles for each expression and then determine their coterminal angles by taking the result modulo 360°.
1. [tex]\( 300^{\circ} - 860^{\circ} \)[/tex]
[tex]\[
300 - 860 = -560 \quad \text{(To find its coterminal angle in the range [0°, 360°], add 360° repeatedly until the result is in this range)} \\
-560 + 360 + 360 = 160
\][/tex]
The coterminal angle for [tex]\( 300^{\circ} - 860^{\circ} \)[/tex] is [tex]\( 160^{\circ} \)[/tex].
2. [tex]\( 300^{\circ} - 840^{\circ} \)[/tex]
[tex]\[
300 - 840 = -540 \quad \text{(To find its coterminal angle in the range [0°, 360°], add 360° repeatedly until the result is in this range)} \\
-540 + 360 + 360 = 180
\][/tex]
The coterminal angle for [tex]\( 300^{\circ} - 840^{\circ} \)[/tex] is [tex]\( 180^{\circ} \)[/tex].
3. [tex]\( 300^{\circ} - 740^{\circ} \)[/tex]
[tex]\[
300 - 740 = -440 \quad \text{(To find its coterminal angle in the range [0°, 360°], add 360° repeatedly until the result is in this range)} \\
-440 + 360 + 360 = 280
\][/tex]
The coterminal angle for [tex]\( 300^{\circ} - 740^{\circ} \)[/tex] is [tex]\( 280^{\circ} \)[/tex].
4. [tex]\( 300^{\circ} - 720^{\circ} \)[/tex]
[tex]\[
300 - 720 = -420 \quad \text{(To find its coterminal angle in the range [0°, 360°], add 360° repeatedly until the result is in this range)} \\
-420 + 360 + 360 = 300
\][/tex]
The coterminal angle for [tex]\( 300^{\circ} - 720^{\circ} \)[/tex] is [tex]\( 300^{\circ} \)[/tex].
Hence, the measures of angles that are coterminal with [tex]\( 300^{\circ} \)[/tex] from the given options are:
1. [tex]\( 160^{\circ} \)[/tex] from [tex]\( 300^{\circ} - 860^{\circ} \)[/tex]
2. [tex]\( 180^{\circ} \)[/tex] from [tex]\( 300^{\circ} - 840^{\circ} \)[/tex]
3. [tex]\( 280^{\circ} \)[/tex] from [tex]\( 300^{\circ} - 740^{\circ} \)[/tex]
4. [tex]\( 300^{\circ} \)[/tex] from [tex]\( 300^{\circ} - 720^{\circ} \)[/tex]
All four expressions provided yield angles that are coterminal with [tex]\( 300^{\circ} \)[/tex].
