To determine the value of [tex]\( A \)[/tex] given that [tex]\( \sin A = \frac{\sqrt{3}}{2} \)[/tex], we need to consider the standard positions on the unit circle where the sine of an angle equals [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
1. Recognize that [tex]\( \sin A = \frac{\sqrt{3}}{2} \)[/tex] is a known value, corresponding famously to [tex]\( 60^\circ \)[/tex] in the unit circle.
2. Thus, one possible value for [tex]\( A \)[/tex] is [tex]\( 60^\circ \)[/tex].
3. However, the sine function is positive in both the first and second quadrants. We must also find the value of [tex]\( A \)[/tex] in the second quadrant.
4. In the second quadrant, the sine of an angle equals [tex]\( \frac{\sqrt{3}}{2} \)[/tex] when the angle is [tex]\( 180^\circ - 60^\circ \)[/tex].
5. Therefore, the second possible value for [tex]\( A \)[/tex] is [tex]\( 180^\circ - 60^\circ = 120^\circ \)[/tex].
To summarize, the two possible values of [tex]\( A \)[/tex], in degrees, such that [tex]\( \sin A = \frac{\sqrt{3}}{2} \)[/tex] are:
[tex]\[ A_1 = 59.99999999999999^\circ \][/tex]
[tex]\[ A_2 = 120.0^\circ \][/tex]
These values are approximately [tex]\( 60^\circ \)[/tex] and exactly [tex]\( 120^\circ \)[/tex].
