Question

Find the exact value of the given expression. $$ \sin ^{-1} 0 $$

Short answer

The exact value of the given expression \(\sin^{-1}(0)\) is 0.

Step-by-step solution

Arcsine Definition

The inverse sine function, denoted by \(\sin^{-1}(x)\) or \(\arcsin(x)\), is the function that, given a value x in the range -1 to 1, returns the angle θ in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that \(\sin(\theta) = x\). In other words, \(\sin^{-1}(0)\) returns an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) whose sine is equal to 0.

Recall the Sine Function from the Unit Circle

On the unit circle, the sine function represents the y-coordinate value of a point, associated with an angle θ. That is, if we know the angle θ, we can calculate the sine value (the y-coordinate) of the point where the terminal side of the angle intersects the unit circle.

Identify the Angle

Now, we have to find the angle θ in the range of \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that \(\sin(\theta) = 0\). Let's take a look at the unit circle: - When θ = 0, the point where the terminal side of the angle intersects the unit circle is (1,0), so \(\sin(\theta) = 0\). - Other angles in the range may not give a sine value of 0. So, \(\sin^{-1}(0) = 0\).

Conclusion

The exact value of the given expression, \(\sin^{-1}(0)\), is 0.