Answer:
1. 60° (or π/3 radians)
2. 120° (or 2π/3 radians)
Step-by-step explanation:
To evaluate [tex]sin^{-1} (\frac{3\sqrt{2} }{2})[/tex] for angles between 0° and 360°, we need to find all angles in this range whose sine equals [tex]\frac{3\sqrt{2} }{2}[/tex]. Let's approach this step-by-step:
1. First, let's recognize that [tex]\frac{3\sqrt{2} }{2}[/tex] ≈ 0.866, which is the sine of a common angle.
2. The principal value of [tex]sin^{-1} (\frac{3\sqrt{2} }{2})[/tex] is 60° or π/3 radians. This is our first solution.
3. However, we need to find all solutions in the range 0° to 360°. The sine function repeats every 360°, and within each 360° cycle, there are two angles with the same sine value (except for 0°, 90°, 180°, 270°, and 360°).
4. To find the second angle in the 0° to 360° range, we can use the property that [tex]sin(180 - ()) = sin()[/tex]. So our second solution is:
[tex]180 - 60 = 120[/tex]
Therefore, the solutions for [tex]sin^{-1} (\frac{3\sqrt{2} }{2})[/tex] between 0° and 360° are:
1. 60° (or π/3 radians)
2. 120° (or 2π/3 radians)
These angles correspond to the points where the sine curve intersects the line [tex]y=\frac{3\sqrt{2} }{2}[/tex] in the first and second quadrants of the unit circle.
