Answer :
To determine the two angles between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] that have a sine of [tex]\(\frac{\sqrt{3}}{2}\)[/tex], we can follow these steps:
1. Identify Key Sine Values: Recall the key angles and their corresponding sine values in the unit circle:
- [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex]
- [tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sin 90^\circ = 1\)[/tex]
2. Match the Given Sine Value: Given that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex], we identify that one of the angles with this sine value is:
[tex]\[ \theta = 60^\circ \][/tex]
3. Identify the Quadrants: The sine function is positive in the first and second quadrants. Hence, we need to find another angle in the second quadrant that has the same sine value.
4. Determine the Second Angle: In the second quadrant, the angle can be found using the supplement of the first quadrant angle:
[tex]\[ \theta = 180^\circ - 60^\circ = 120^\circ \][/tex]
Therefore, the two angles between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] that have a sine of [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are:
[tex]\[ \boxed{60^\circ \text{ and } 120^\circ} \][/tex]
1. Identify Key Sine Values: Recall the key angles and their corresponding sine values in the unit circle:
- [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex]
- [tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sin 90^\circ = 1\)[/tex]
2. Match the Given Sine Value: Given that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex], we identify that one of the angles with this sine value is:
[tex]\[ \theta = 60^\circ \][/tex]
3. Identify the Quadrants: The sine function is positive in the first and second quadrants. Hence, we need to find another angle in the second quadrant that has the same sine value.
4. Determine the Second Angle: In the second quadrant, the angle can be found using the supplement of the first quadrant angle:
[tex]\[ \theta = 180^\circ - 60^\circ = 120^\circ \][/tex]
Therefore, the two angles between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex] that have a sine of [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are:
[tex]\[ \boxed{60^\circ \text{ and } 120^\circ} \][/tex]
