91Ó°ÊÓ

Inverse trigonometric functions are functions that reverse the action of the trigonometric functions. They help find the angles when the values of the trigonometric ratios are known. For example, \(\tan^{-1}(y)\) finds the angle whose tangent is y.To understand the exercise, it's crucial to recall that \(\tan^{-1}(-1)\) means we need the angle whose tangent is -1. All inverse trigonometric functions have specific ranges for their outputs:
- \(\tan^{-1}\theta\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)
- \(\text{One must be mindful of these ranges when dealing with these functions to ensure correctness. Furthermore, knowing which quadrant the angles fall into based on the signs of the outputs aids in correctly determining these values.\)

The tangent function is one of the main trigonometric functions and is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. Simply put, \(\tan(\theta)\) = opposite/adjacent. If we look at the unity circle, tangent represents the slope of the line that forms the angle \(\theta\). For example, In our exercise, \(\tan^{-1}(-1)\), the tangent of \(\theta\) is -1. This means that the angle where the ratio of the opposite to the adjacent side is -1 lies in the fourth quadrant.
- Because the tangent is negative, the angle is in the fourth quadrant.
- It results in \(\theta = -\frac{\pi}{4}\). When dealing with tangent inverses, it’s always helpful to remember they represent angles where their tangents equate to the provided number.

The sine function, another fundamental trigonometric function, is the ratio of the opposite side to the hypotenuse in a right-angled triangle. Mathematically, \(\textsin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). When translating to the unit circle, sine refers to the Y-coordinate of the point where the terminal side of the angle \(\theta\) intersects the unit circle.
For this exercise, once we find that \(\theta = -\frac{\pi}{4}\), we need to calculate the sine of this angle:

\(\sin(\theta) = \sin\left(-\frac{\pi}{4}\right)\)

Using symmetry properties of the unit circle, sine of negative angles is the negative of the sine of positive ones: \(\sin\left(-\frac{\text{\pi}}{4}\right) = -\sin\left( \frac{\text{\pi}}{4}\right)\)

Knowing \(\sin\left(\frac{\text{\pi}}{4}\right) = \frac{\sqrt{2}}{2}\), we conclude: \(\sin\left(-\frac{\text{\pi}}{4}\right) = -\frac{\sqrt{2}}{2}\)\