Problem 55 - Calculus

Question

This problem introduces the arccosine function, or inverse cosine, denoted by \(\left[\mathrm{cos}^{-1}\right]\) on most calculators. (a) Using a calculator set in radians, make a table of values, to two decimal places, of \(g(x)=\arccos x,\) for \(x=-1,-0.8,-0.6, \ldots, 0, \ldots, 0.6,0.8,1\) (b) Sketch the graph of \(g(x)=\arccos x\) (c) Why is the domain of the arccosine the same as the domain of the arcsine? (d) What is the range of the arccosine? (e) Why is the range of the arccosine not the same as the range of the arcsine?

Step-by-Step Solution

Answer

(a) Calculated values, (b) graph sketched, (c) same domain as arcsine to fit cosine range, (d) range [0, π], (e) ranges differ due to angle interpretations.

Step 1: Calculate Values for g(x) = arccos(x) using a Calculator

Using a calculator set in radians mode, compute the value of \( g(x) = \arccos(x) \) for each specified value of \( x \). Record the results to two decimal places.- \( x = -1 \), \( g(x) = \arccos(-1) = \pi \) which is approximately 3.14- \( x = -0.8 \), \( g(x) = \arccos(-0.8) \approx 2.50 \)- \( x = -0.6 \), \( g(x) = \arccos(-0.6) \approx 2.21 \)- \( x = -0.4 \), \( g(x) = \arccos(-0.4) \approx 1.98 \)- \( x = -0.2 \), \( g(x) = \arccos(-0.2) \approx 1.77 \)- \( x = 0 \), \( g(x) = \arccos(0) = \frac{\pi}{2} \) which is approximately 1.57- \( x = 0.2 \), \( g(x) = \arccos(0.2) \approx 1.37 \)- \( x = 0.4 \), \( g(x) = \arccos(0.4) \approx 1.16 \)- \( x = 0.6 \), \( g(x) = \arccos(0.6) \approx 0.93 \)- \( x = 0.8 \), \( g(x) = \arccos(0.8) \approx 0.64 \)- \( x = 1 \), \( g(x) = \arccos(1) = 0 \).

Step 2: Sketch the Graph of g(x) = arccos(x)

Plot the computed values from Step 1 on a Cartesian plane where the x-axis represents the input values of \(x\) and the y-axis represents the corresponding \(g(x) = \arccos(x)\) values. Connect these points with a smooth curve that represents the inverse cosine function from \(-1\) to \(1\). The graph should start at the point \((-1, \pi)\), dip down to \((0, \frac{\pi}{2})\), and end at \((1, 0)\).

Step 3: Explain the Domain of the Arccosine Function

The domain of \(\arccos(x)\) is \([-1, 1]\) because the cosine function, whose inverse is the arccosine, takes values between \(-1\) and \(1\). This is why this range is also the domain for both \(\arcsin(x)\) and \(\arccos(x)\).

Step 4: Determine the Range of the Arccosine Function

The range of \(\arccos(x)\) is \([0, \pi]\). This means \(\arccos(x)\) will output values within this interval as the cosine of any angle in this range will be within \([-1, 1]\).

Step 5: Compare Ranges of Arccosine and Arcsine Functions

The range of \(\arccos(x)\) is \([0, \pi]\) while the range of \(\arcsin(x)\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). The ranges are different because \(\arccos(x)\) resolves angles from the x-axis upward (first and second quadrants), whereas \(\arcsin(x)\) resolves from the x-axis both upward and downward (first and fourth quadrants) to capture negative input values.

Key Concepts

Arccosine Function

The arccosine function, often written as \( \arccos(x) \), is the inverse of the cosine function. This means that the arccosine provides the angle for which the cosine is a given value.

• The arccosine function is defined for all real numbers \( x \) where \(-1 \leq x \leq 1\).
• Using this function, we can determine the angle \( \theta \) such that \( \cos(\theta) = x \).
• On a scientific calculator, this function is commonly labeled as \( \cos^{-1}(x) \).

This inverse function is particularly useful in trigonometry when we know the cosine of an angle but need to find the actual angle in radians. Remember that the outputs of the arccosine function provide us with angles between \( 0 \) and \( \pi \) radians.

Function Domain

The domain of a function represents all possible input values that the function can accept. In the case of the arccosine function, the domain is determined by the values the original cosine function can output.
For \( \arccos(x) \):

• The domain is \([-1, 1]\).
• This means that any input value for \( x \) must lie between \(-1\) and \( 1\), inclusive.

This domain corresponds to the possible values of the cosine function, as cosine of any angle can only range from \(-1\) to \( 1\). The domain of the arccosine is the same as that of the arcsine function since both are based on the unit circle properties and their originating trigonometric functions share this range.

Function Range

The range of the arccosine function \( \arccos(x) \) tells us what values the output (or angle) will take when the input values lie within the domain.

• For \( \arccos(x) \), the range is \([0, \pi]\).
• This means the output of the function spans all angles from \(0\) to \(\pi\) radians.

This range is derived because the arccosine function captures angles only in the first and second quadrants of the unit circle. Comparing this with the range of the arcsine function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), we can see different intervals since arcsine outputs angles from both the upward and downward directions from the x-axis.

Graph Sketching

Sketching the graph of the \( \arccos(x) \) function helps visualize how it behaves across its domain.
To sketch the graph:

• Plot points starting at \((-1, \pi)\) and ending at \((1, 0)\).
• The graph should pass smoothly through intermediate points calculated, like \((0, \frac{\pi}{2})\) or any other computed values such as \((-0.6, 2.21)\).
• Ensure the curve starts high and descends steadily as \(x\) increases from \(-1\) to \(1\).

Practicing graph sketching reinforces understanding of the function's behavior over its domain and range. It also highlights the inverse relationship with the cosine's periodic nature.