How can cosx = x have more than one solution?

  • Thread starter Jan Hill
  • Start date
In summary, the conversation discusses the question of how cosx = x can have more than one solution. The participant initially believes that the wave nature of the graph of cosx may be the reason, but upon examining the graphs of y = cosx and y = x, it is found that they only intersect at one point. The conversation then explores the possibility of the gradient of cosx being the limiting factor for the number of solutions. Ultimately, it is concluded that there is only one solution for cosx = x.
  • #1
Jan Hill
63
0

Homework Statement



How can cosx = x have more than one solution?

Homework Equations





The Attempt at a Solution

I don't know how to begin this.
 
Physics news on Phys.org
  • #3


because of the wave nature of the graph of cosx
 
  • #4


Sketch graphs of y = cos(x) and y = x. At how many points do these graphs intersect?

As an alternative you can graph y = cos(x) - x. The x-intercepts of this graph indicate the values for which cos(x) = x.
 
  • #5


the graphs intersect at more than 1 point, in fact at 3 points and does that mean we can say that that is why cosx = x has more than one solution because is more than one value for x at which cos x equals it?
 
  • #6


The only intersection point I see is for x ~ .72. What did you graph?
 
  • #7


I don't understand how that helps us get further with cosx = x having more than one solution
 
  • #8


My earlier question was: why do you think there are more than one solution? You said the the graphs of y = cosx and y = x intersect at three points. My response is that the only intersection point I find is when x is about .72.

I don't believe there are any other points of intersection. If you believe there are, what are the x values at these points?
 
  • #9


I initially looked for more than one solution for cosx = x because that's the way the question was posed but I see now that it was just to show that in fact, there is only one solution to it and that, as seen by looking at the graphs is x is about .72

Thank you for your help
 
  • #10


The gradient of cosx is never more than 1 so a line of gradient 1 or more can only ever cut y=cosx once.
 
  • #11


Got it. thanks
 

FAQ: How can cosx = x have more than one solution?

How can a single equation have more than one solution?

The equation cosx = x is a trigonometric equation that represents the intersection points of a cosine function and a line. Since the cosine function is a periodic function with a period of 2π, there can be multiple values of x that satisfy the equation at different points on the graph.

Can you give an example of how cosx = x can have more than one solution?

One example of an x value that satisfies cosx = x is 0, since cos0 = 1 and 0 = 0. However, another value that satisfies the equation is approximately 0.739, since cos0.739 ≈ 0.739. This value is also known as the first positive root of the equation.

How can I find all the solutions to cosx = x?

To find all the solutions to cosx = x, you can use a graphing calculator or software to plot the cosine function and the line y = x. The intersection points of these two graphs will give you the solutions to the equation. Additionally, you can use algebraic methods such as factoring or using the unit circle to find other solutions.

Are all the solutions to cosx = x real numbers?

Yes, all the solutions to cosx = x are real numbers. This is because both the cosine function and the line y = x have a domain and range of all real numbers, meaning that any value of x that satisfies the equation will be a real number.

Why is it important to understand that cosx = x can have more than one solution?

Understanding that cosx = x can have more than one solution is important because it allows us to accurately solve and interpret trigonometric equations. It also helps us understand the behavior of trigonometric functions and their relationship with other functions. Additionally, this concept is important in various applications of trigonometry, such as in physics and engineering.

Similar threads

Integrating 1/(1+cosx) from -pi/2 to pi/2
Replies
5
Views
1K
Integration problem ∫sqroot((cosx-cos^3(x))/(1-cos^3(x)))
Replies
18
Views
2K
Derivative of a sinusoidal function
Replies
2
Views
1K
How can the inequality cosx ≥ (1-x^2/2) be proven?
Replies
9
Views
2K
Differentiation of Log(cos(X)/x^2)^2
Replies
6
Views
1K
Method of Undetermined Coefficients
Replies
1
Views
2K
Area under a curve-positive and negative area cancellation
Replies
2
Views
2K
Find a particular solution of y"+2y'+y=8x^2*cosx-4xsinx
Replies
11
Views
3K
How to Solve a Second Order Nonhomogeneous Differential Equation?
Replies
9
Views
2K
Wondering if these two First Linear Order IVPs are correct
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top