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Magazine Chapter 6: Problem 67
Use the graphs of \(y=\tan x\) and \(y=\cos x+1\) to determine the number of solutions to the equation \(\tan x=\cos x+1\) on the interval \([0,2 \pi)\).
Short Answer
Expert verified
There are 4 solutions in the interval \([0, 2\pi)\).
Step by step solution
01
Understand the Problem
We need to find the number of solutions for the equation \(\tan x = \cos x + 1\) within the interval \([0, 2\pi)\). This requires analyzing the graphs of \(y = \tan x\) and \(y = \cos x + 1\).
02
Analyze the Graph of \(\tan x\)
The graph of \(y = \tan x\) has vertical asymptotes at \(x = \frac{\pi}{2}, \frac{3\pi}{2}\), and it resets after every \(\pi\) period. It increases from \(-\infty\) to \(\infty\) between these asymptotes.
03
Analyze the Graph of \(\cos x+1\)
The function \(y = \cos x + 1\) oscillates between 0 and 2. It is a vertical translation of \(y = \cos x\) upwards by 1 unit, covering complete cycles at multiples of \(2\pi\).
04
Compare the Graphs
Now we compare the graphs between \([0, 2\pi)\). In this range, the \(\tan x\) graph has asymptotes at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\). The \(\cos x + 1\) graph intersects the \(\tan x\) graph where their values are equal.
05
Count Intersections
In the interval between each of the \(\tan x\) asymptotes, \(\tan x\) increases from \(-\infty\) to \(\infty\), crossing the \(\cos x + 1\) curve exactly once in each interval. Thus, there are two intersections in the interval \((0, \pi)\) and two more in \((\pi, 2\pi)\).
06
Verify the Solution
Each intersection corresponds to a solution for the equation \(\tan x = \cos x + 1\). Therefore, the entire interval \([0, 2\pi)\) will have 4 solutions in total to the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graph Analysis
Graph analysis is a powerful tool to visually identify solutions to trigonometric equations. To solve the equation \( \tan x = \cos x + 1 \) in the interval \([0, 2\pi)\), we need to consider the behavior and characteristics of the graphs of \(y = \tan x\) and \(y = \cos x + 1\).
- The graph of \(y = \tan x\) has vertical asymptotes at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\). This means that as \(x\) approaches these points, the function's value tends to infinity or negative infinity.
- Similarly, the graph of \(y = \cos x + 1\) is a vertical shift of \(y = \cos x\) by 1 unit upwards, oscillating between 0 and 2.
Tangent Function
The tangent function, represented by \(y = \tan x\), is an essential trigonometric function crucial to solving many equations. Understanding its main properties helps in determining where it overlaps with other graphs, like \(y = \cos x + 1\).
- \(\tan x\) has a period of \(\pi\), meaning it repeats every \(\pi\) units. Unlike the \(\cos x\) or \(\sin x\) functions, which repeat every \(2\pi\), \(\tan x\) resets more frequently.
- Between each period, \(\tan x\) starts at \(-\infty\), approaches 0, and then increases towards \(+\infty\).
- The vertical asymptotes at \(x = \frac{\pi}{2}\), \(\frac{3\pi}{2}\) divide the graph into sections where this behavior occurs.
Cosine Function
The cosine function, specifically in the form \(y = \cos x + 1\), is a shifted version of the commonly known \(y = \cos x\). Understanding this vertical shift is vital when analyzing the intersection points with other trigonometric graphs.
- The graph of \(y = \cos x + 1\) means that the entire cosine wave is moved one unit up, resulting in a range from 0 to 2, compared to the standard cosine wave from -1 to 1.
- Despite this shift, the periodic nature of the cosine function remains unchanged, with a cycle every \(2\pi\).
- This constant upward shift ensures that within any complete cycle from 0 to \(2\pi\), the graph consistently hovers above the x-axis, influencing where and how it intersects with the \(\tan x\) curve.
