Questions about these Trigonometry Graphs involving sin() and cos()

In summary, the content discusses various questions related to the graphs of sine and cosine functions in trigonometry. It explores key characteristics such as amplitude, period, phase shift, and vertical shift, and how these attributes affect the shape and position of the graphs. Additionally, it may address transformations of the basic sine and cosine graphs and their applications in solving trigonometric problems.
  • #1
pairofstrings
411
7
TL;DR Summary
a sin(x) - b cos(y) = y
a sin(x) + b cos(y) = 1
Hi.
I have two trigonometric equations whose graphs I am trying to understand.
Here are the equations:
1. a sin(x) - b cos(y) = y; a = 2, b = 2

Web capture_20-8-2023_152359_www.desmos.com.jpeg

2. a sin(x) + b cos(y) = 1; a = 1, b = 1

Web capture_20-8-2023_15261_www.desmos.com.jpeg

My question is why the graphs are the way they are.
What should I do to understand them?
Can anyone explain these graphs?

Thanks for the help.
 
Last edited:
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  • #2
When you consider level sets ##\{(x,y)\mid f(x,y)=const\}## it is important to find critical points of the function ##f## and understand which kind these critical points are.
So first find the points such that ##df=0##.
It is like drawing a phase portrait of a Hamiltonian system with the Hamiltonian f.
 
Last edited:
  • #3
Thanks. So, I need to do Analysis first?
 
  • #4
pairofstrings said:
Thanks. So, I need to do Analysis first?
The second graph looks off to me. You have
$$\cos y = 1 - \sin x$$If ##\sin x <0##, then there are no solutions for ##y##. You have solutions for ##0 \le x \le \pi##, with symmetry about ##x = \frac \pi 2##. Whatever solutions you have in this range are repeated every ##2\pi## units along the x-axis.

It would be better have units of ##\pi## along both axes.

Does that get you started?
 

FAQ: Questions about these Trigonometry Graphs involving sin() and cos()

What is the general shape of the sine and cosine graphs?

The sine and cosine graphs are both periodic waveforms that oscillate between -1 and 1. The sine graph starts at the origin (0,0) and rises to its maximum value of 1 at π/2, then decreases back to 0 at π, continues to -1 at 3π/2, and returns to 0 at 2π. The cosine graph starts at its maximum value of 1 when x = 0, decreases to 0 at π/2, reaches -1 at π, returns to 0 at 3π/2, and goes back to 1 at 2π.

What are the key differences between the sine and cosine functions?

The key difference between sine and cosine functions lies in their phase shift. The sine function is defined as the y-coordinate of a point on the unit circle, while the cosine function is defined as the x-coordinate. Consequently, the sine graph is a phase shift of the cosine graph to the right by π/2. In terms of their equations, sin(x) and cos(x) have the same amplitude and period but differ in their starting points.

How do the amplitude and period of sine and cosine functions affect their graphs?

The amplitude of both sine and cosine functions determines the height of the peaks and the depth of the troughs of the graphs. For a function of the form y = A sin(Bx) or y = A cos(Bx), the amplitude is given by |A|, while the period is calculated as (2π)/|B|. A larger amplitude results in taller peaks and deeper troughs, while a smaller amplitude compresses the graph vertically. The period affects how frequently the graph oscillates; a larger value of B results in more cycles within a given interval.

How do phase shifts affect the sine and cosine graphs?

A phase shift occurs when the graph of a sine or cosine function is horizontally shifted along the x-axis. For the functions of the form y = sin(B(x - C)) or y = cos(B(x - C)), the phase shift is given by C. A positive value of C shifts the graph to the right, while a negative value shifts it to the left. This affects where the peaks, troughs, and intercepts occur on the graph, but does not change the amplitude or period of the function.

What are the applications of sine and cosine graphs in real life?

Sine and cosine graphs have numerous applications in real life, particularly in fields such as physics, engineering, and signal processing. They are used to model periodic phenomena such as sound waves, light waves, and vibrations. For instance, they can represent the motion of pendulums, the

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