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Chapter 4: Problem 83

Sketch a graph of the function. \(y=2 \arccos x\)

Short Answer

Expert verified
The graph of the function is a decreasing curve, starting from the point \((1,0)\), going through \((0, \pi)\), and ending at \((-1, 2\pi)\). It is limited to the domain [-1,1].

Step by step solution

01

Understanding the base function

Firstly, consider the basic function \(y = \arccos x\) . The arccosine function is the inverse of the cosine function. It has a domain of [-1,1] and a range of [0, \(\pi\)]. It is decreasing over its domain and it crosses the y-axis at \((0, \pi/2)\).
02

Sketching the base function

To understand the impact of multiplying the arccosine function by 2, start by sketching the graph of \(y = \arccos x\). Draw a decreasing curve, starting from the point \((1,0)\), going through \((0, \pi/2)\), and ending at \((-1, \pi)\).
03

Modifying the base function

The given function \(y=2 \arccos x\) is a vertical stretch of the base function by a factor of 2. To sketch this, the y-coordinate of each point should be multiplied by 2. So, the graph still starts at \((1,0)\), it crosses the y-axis at \((0, \pi)\), and ends at \((-1, 2\pi)\). It is still decreasing over the interval from -1 to 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Trigonometric Functions
Inverse trigonometric functions are mathematical functions that reverse the action of the standard trigonometric functions, such as sine, cosine, and tangent. When we have a ratio of sides in a right-angled triangle, trigonometric functions allow us to find the angles, whereas their inverses do the opposite: given an angle, we determine the ratio of sides.

Specifically, the arccosine, denoted as \(\arccos\), is the inverse of the cosine function. Here鈥檚 what this means: for any number \(x\) within the domain of \(\cos\), \(\arccos\) answers the question, \

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Most popular questions from this chapter

Use a graphing utility to graph \(f\) \(g,\) and \(y=x\) in the same viewing window to verify geometrically that \(g\) is the inverse function of \(f .\) (Be sure to restrict the domain of \(f\) properly.) \(f(x)=\tan x, \quad g(x)=\arctan x\)

Evaluate the expression without using a calculator. \(\arctan \frac{\sqrt{3}}{3}\)

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x .\) \(g(x)=\arcsin \frac{x}{2}\)

Use the properties of inverse trigonometric functions to evaluate the expression. \(\tan (\arctan 45)\)

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\sin \left(\arctan \frac{3}{4}\right)\)
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