Problem 106 - Precalculus

Question

Why do the functions \(f(x)=\sin ^{-1} x\) and \(g(x)=\cos ^{-1} x\) have different ranges?

Step-by-Step Solution

Answer

Their ranges differ due to the intervals where each function is one-to-one on the unit circle.

Step 1: Understand the Definitions

The function \(f(x)=\sin^{-1}x\) represents the inverse sine function (also called arcsin), which returns the angle whose sine is \(x\). The function \(g(x)=\cos^{-1}x\) represents the inverse cosine function (also called arccos), which returns the angle whose cosine is \(x\).

Step 2: Determine the Range of \(f(x) = \sin^{-1}x\)

The range of the inverse sine function \(\sin^{-1}x\) is the set of possible output angles. \(\sin^{-1}x\) outputs angles from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), inclusive. This is because these angles correspond to the interval where the sine function is one-to-one.

Step 3: Determine the Range of \(g(x) = \cos^{-1}x\)

The range of the inverse cosine function \(\cos^{-1}x\) is the set of possible output angles. \(\cos^{-1}x\) outputs angles from \(0\) to \(\pi\) because the cosine function is one-to-one in this interval.

Step 4: Compare the Ranges

The range of \(f(x) = \sin^{-1}x\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), while the range of \(g(x) = \cos^{-1}x\) is \([0, \pi]\). These ranges differ because the sine and cosine functions are one-to-one over different intervals on the unit circle.

Key Concepts

range of inverse functions

Inverse functions reverse or "undo" the process of the original trigonometric functions. Understanding the range of inverse trigonometric functions like arcsine \( \sin^{-1} x \) (also called \( \arcsin(x) \)) and arccosine \( \cos^{-1} x \) (or \( \arccos(x) \)) is essential because it defines the set of output values that the function can produce.

• For a function to have an inverse, it must be one-to-one (bijective), meaning it needs to pass both the horizontal and vertical line tests.
• Due to this constraint, we must restrict the domains of trigonometric functions when considering their inverses.
• These restrictions determine which angles form the basis of the ranges of the inverse functions.

The range of an inverse function is derived from the domain of the original trigonometric function that has been restricted to maintain a one-to-one relationship. This is why the ranges of \( \arcsin \) and \( \arccos \) are different, reflecting their respective restricted domains.

arcsin function

The inverse sine function, often written as \( \arcsin \) or \( \sin^{-1} \), finds the angle whose sine value is given. This angle is expressed in radians.

• Range: The range of \( \arcsin(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• This means \( \arcsin(x) \) produces angles from \( -\frac{\pi}{2} \) (or \(-90°\)) to \(\frac{\pi}{2} \) (or \(90°\)).

These values ensure the function remains one-to-one.
On the unit circle, this corresponds to the right half of the circle, covering the first and fourth quadrants.

Why This Range?

The range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) is chosen because the sine function is one-to-one and non-repeating on this interval.
Therefore, every value in this range corresponds to exactly one angle where sine achieves each value between \(-1\) and \(1\).

arccos function

The inverse cosine function, represented by \( \arccos \) or \( \cos^{-1} \), calculates the angle whose cosine is a specific value. Like arcsine, it outputs angles in radians.

• Range: For \( \arccos(x) \), the range is \([0, \pi]\).
• This indicates \( \arccos(x) \) produces angles ranging from \(0\) (or \(0°\)) to \( \pi \) (or \(180°\)).

These angles cover the upper half of the unit circle, which includes the first and second quadrants.

Why These Ranges Matter?

The choice of range for \( \arccos(x) \) ensures the cosine function remains one-to-one.
Within \([0, \pi]\), each cosine value corresponds to exactly one angle on this part of the circle. This guarantees a unique output, vital for defining the behavior of inverse functions.
Hence, it differs from the arcsin function due to the behavior and symmetry of the cosine wave on the unit circle.