*HELP*Investigate Values of m in lR where cosx=mx has 2 Solutions

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SUMMARY

The discussion focuses on solving the equation cos(x) = mx by analyzing the intersections of the graphs y = cos(x) and y = mx for specific values of m, namely m = 1 and m = 1/5. Participants are tasked with sketching these graphs to visually determine the number of solutions. It is established that for certain values of m, specifically -m, the equation will also yield exactly two solutions. The discussion raises the question of whether values of m can yield three or four solutions, prompting further graphical analysis.

PREREQUISITES
  • Understanding of trigonometric functions, specifically cosine.
  • Graphing skills for functions in Cartesian coordinates.
  • Knowledge of linear equations and their graphical representations.
  • Familiarity with the concept of intersections in graph analysis.
NEXT STEPS
  • Explore the behavior of trigonometric functions and their intersections with linear functions.
  • Investigate the implications of varying the slope (m) in the equation cos(x) = mx.
  • Learn about graphical methods for estimating solutions to equations.
  • Study the conditions under which trigonometric equations can have multiple solutions.
USEFUL FOR

Students studying calculus, particularly those focusing on trigonometric equations, as well as educators seeking to enhance their teaching methods for graphing and solution estimation techniques.

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Homework Statement



(a)Carefully draw graphs of y=cosx, y=x, y=(1/5)x
(b) From the graphs, determine how many solutions the equation cosx=mx has for m=1 and m=1/5
(c) Add a sketch of y=mx onto your picture, where m is chosen so that cosx=mx has exactly two solutions. (without computing the value of m)
(d) Use your sketch to estimate the correct value of m
(e) why is it true that the value of -m will also give exactly two solutions to cosx=mx?
(f)Is it possible to find a value of m such that cosx=mx has exactly three solutions? four? n?
Explain by sketching y=cosx and y=mx for relevant values of m

The Attempt at a Solution


A hint was given to us by the prof:
"you should draw all your graphs on just one diagram like: (the attachment)"

Okay, so I don't have a problem drawing the graphs,but for part (b):confused: and rest:eek:
Please Help:redface:
 

Attachments

  • Picture 1.png
    Picture 1.png
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If you are able to do a, why does your 'attempt at a solution' not show that at least?
If you can in fact draw the graph in (a) all you have to do to answer (b) is look at that graph and count the number of times one graph crosses the other.
 

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