Understanding the Transition to Solve for t in an Oscillation Function

In summary, the cosine function can be used to model oscillation with the equation x=A cos(wt). To solve for t, the inverse function of cosine, arccos, is used. This results in the equation t= (arccos (x/a)/2pi)*T, where 2pi and T come from w=2pi/T. To find the value of arccos(cos(wt)), the equation is divided by A and then the arccos is taken on both sides.
  • #1
Mynona
3
0
I have a cosine function, namely the function for oscillation x=A cos(wt).

I want to separate the t out here, so I can solve for it. My teacher gave the the answer to be t= (arccos (x/a)/2pi)*T, but I can't quite see where he came up with that. Would anyone be as kind as to give me a more elaborate explanation of how this transition was made.


(2pi and T comes from w=2pi/T).

Sorry for bad English, not a native English speaker obviously :)
 
Mathematics news on Phys.org
  • #2
The inverse function of the cosine is the arc cosine. If a function [itex]f(x)[/itex] has an inverse [itex]f^{-1}(x)[/itex] then [itex]f^{-1}(f(x))=f(f^{-1}(x))=x[/itex].

Divide the equation on both sides by A then take the arccos on both sides. Can you calculate [itex]\arccos(\cos(\omega t)) [/itex]now?
 

1. What is the inverse cosine function?

The inverse cosine function, also known as arccosine, is the inverse operation of the cosine function. It is denoted as cos-1 or arccos and it takes the value of an angle in radians as its input and gives the ratio of the adjacent side to the hypotenuse of a right triangle as its output.

2. What is the domain and range of the inverse cosine function?

The domain of the inverse cosine function is the set of real numbers between -1 and 1, inclusive, since the cosine function can only take values between -1 and 1. The range of the inverse cosine function is the set of all real numbers between 0 and π radians, or 0 and 180 degrees, inclusive.

3. How is the inverse cosine function related to the cosine function?

The inverse cosine function is the inverse operation of the cosine function. This means that if an angle θ is put into the cosine function, the output will be the cosine of that angle. Conversely, if the cosine of an angle x is put into the inverse cosine function, the output will be the angle x.

4. What is the graph of the inverse cosine function?

The graph of the inverse cosine function is a reflection of the graph of the cosine function about the line y=x. This means that the x and y coordinates of the points on the graph are swapped. The graph of the inverse cosine function is a curve that starts at (1,0) and approaches (0,π/2) as the input approaches -1, and approaches (0,π) as the input approaches 1.

5. How is the inverse cosine function used in real life?

The inverse cosine function is used in various fields such as physics, engineering, and mathematics. It is commonly used in trigonometry to solve for missing angles or side lengths in right triangles. It is also used in calculating the phase shift and amplitude in wave functions. Additionally, it has applications in computer graphics and signal processing.

Similar threads

B Sine/Cosine behaving like a linear function
    • General Math
Replies
8
Views
5K
Simple Harmonic motion of a Pendulum
    • Introductory Physics Homework Help
Replies
1
Views
844
MHB Solve Trig Equation Graphically & Algebraically: Why Different Answers?
    • General Math
Replies
11
Views
6K
MHB Inverse Functions and "Verifying"
    • General Math
Replies
1
Views
1K
MATLAB Turning Fourier coefficients into an interpolated freq domain function
    • MATLAB, Maple, Mathematica, LaTeX
Replies
8
Views
1K
I Solving and manipulating the damped oscillator differential equation
    • Differential Equations
Replies
6
Views
2K
B Determining ansatz for forced oscillations in SHM
    • Classical Physics
Replies
1
Views
918
How to find the velocity of a wave in simple harmonic motion given time
    • Introductory Physics Homework Help
Replies
3
Views
860
Simple Harmonic Motion: why sin(wt) instead of sin(t)?
    • Mechanics
Replies
3
Views
5K
Understanding inverse functions
    • General Math
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top