text: # Determine the Equation Of Sinusodial Function
  
  • Oct 19th 2009, 03:39 PM
  mmhaaz
  Determine the Equation Of Sinusodial Function
  Could someone please explain how I would find the ht of sine and cosine on this graph? I don't understand what the difference is. Also how would I find the period? Is it the same for cosine and sine? Thank-you to anyone that could help!! Not in radian form please...
  • Oct 20th 2009, 06:15 AM
  Finding a sine and cosine function for a given graph
  Hello mmhaaz
  Quote:
  
  Originally Posted by mmhaaz
  Could someone please explain how I would find the ht of sine and cosine on this graph? I don't understand what the difference is. Also how would I find the period? Is it the same for cosine and sine? Thank-you to anyone that could help!! Not in radian form please...
  
  First: the difference between the graphs of sine and cosine: there is none, except a 'phase shift' of $\displaystyle 90^o$. In other words, if you shift the cosine graph $\displaystyle 90^o$ to the right, you get the sine graph. See the attached diagram.
  
  Note in particular the 'starting' values; i.e. the values when $\displaystyle x = 0$:
  
  • $\displaystyle \cos 0^o=1$
  • $\displaystyle \sin0^o=0$
  
  Then, the particular questions relating to the graph you've attached.
  
  The scale on the vertical axis isn't very clear, but it looks as if one square = one unit.
  
  So, on that assumption, the graph has a maximum value of $\displaystyle 1$ and a minimum value of $\displaystyle -5$. The sinusoidal axis is the horizontal line that divides the graph exactly in half. So this is the line $\displaystyle y = -2$ (which I think is what you got).
  
  Then, the amplitude is the vertical distance from the sinusoidal axis to one of the peaks or troughs. So that's $\displaystyle 3$ (again, I think you got that).
  
  From these two facts, we can deduce that the equation of the graph will be of the form:
  $\displaystyle y=-2+3\sin$(something) or $\displaystyle y = -2+3\cos$(something)
  The question then refers to the 'starting point', based on a sine and then a cosine graph. I assume that means the point which corresponds to the point where $\displaystyle x = 0$ on the graphs of the basic graphs (the ones in my diagram).
  
  So, for sine, this is the point that corresponds to $\displaystyle (0,0)$ on the basic graph - in other words one of the points where the graph crosses the sinusoidal axis and is moving upwards in a direction from left to right. There's more than one such point, but the obvious one is $\displaystyle (30, -2)$. (Can you see that another one will be $\displaystyle (210,-2)$?)
  
  For the cosine function, the starting point will be the point corresponding to $\displaystyle (0,1)$ on the basic graph. So we're looking for a peak (maximum) value: the obvious one here is $\displaystyle (75,1)$.
  
  The period is the horizontal distance between two consecutive corresponding points on the graph. We said above that two possible starting points for the sine graph are $\displaystyle (30,-2)$ and $\displaystyle (210,-2)$. So the period is the distance between these two points. That's $\displaystyle 180^o$.
  
  Divide by $\displaystyle 360^o$ for the next answer: $\displaystyle \frac{180}{360}=\frac12$.
  
  If we divide $\displaystyle x$ by this fraction ($\displaystyle \tfrac12$) we get $\displaystyle 2x$, which gives us the coefficient of $\displaystyle x$ in the functions we're looking for. (A complete period for $\displaystyle y = \sin x$ is when $\displaystyle x$ goes from $\displaystyle 0^o$ to $\displaystyle 360^o$; so if we want a period of $\displaystyle 180^o$, we shall need values of $\displaystyle 2x$, because they will go from $\displaystyle 0^o$ to $\displaystyle 360^o$ when $\displaystyle x$ goes from $\displaystyle 0^o$ to $\displaystyle 180^o$.)
  
  The 'starting points' that we've worked out above tell us the angle that has to be added or subtracted from $\displaystyle x$ to give the correct starting value - the phase-shift.
  
  So, for the sine function we want to start at $\displaystyle x = 30$, so the function is
  $\displaystyle y=-2+3\sin(2[x-30])$
  And for the cosine function, we want to start at $\displaystyle x = 75$, so the function is
  $\displaystyle y=-2+3\cos(2[x-75])$
explanation: The document is a forum post where a user asks for help in determining the equation of a sinusoidal function from a given graph. The user specifically asks about finding the height (amplitude) of sine and cosine on the graph, the difference between sine and cosine, and the period of the function. Another user provides a detailed response explaining the differences between sine and cosine graphs, how to find the amplitude, the sinusoidal axis, the period, and the phase shift. The response also includes the equations for both the sine and cosine functions based on the given graph.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Determine the Equation Of Sinusodial Function
  
  • Oct 19th 2009, 03:39 PM
  mmhaaz
  Determine the Equation Of Sinusodial Function
  Could someone please explain how I would find the ht of sine and cosine on this graph? I don't understand what the difference is. Also how would I find the period? Is it the same for cosine and sine? Thank-you to anyone that could help!! Not in radian form please...
  • Oct 20th 2009, 06:15 AM
  Finding a sine and cosine function for a given graph
  Hello mmhaaz
  Quote:
  
  Originally Posted by mmhaaz
  Could someone please explain how I would find the ht of sine and cosine on this graph? I don't understand what the difference is. Also how would I find the period? Is it the same for cosine and sine? Thank-you to anyone that could help!! Not in radian form please...
  
  First: the difference between the graphs of sine and cosine: there is none, except a 'phase shift' of $\displaystyle 90^o$. In other words, if you shift the cosine graph $\displaystyle 90^o$ to the right, you get the sine graph. See the attached diagram.
  
  Note in particular the 'starting' values; i.e. the values when $\displaystyle x = 0$:
  
  • $\displaystyle \cos 0^o=1$
  • $\displaystyle \sin0^o=0$
  
  Then, the particular questions relating to the graph you've attached.
  
  The scale on the vertical axis isn't very clear, but it looks as if one square = one unit.
  
  So, on that assumption, the graph has a maximum value of $\displaystyle 1$ and a minimum value of $\displaystyle -5$. The sinusoidal axis is the horizontal line that divides the graph exactly in half. So this is the line $\displaystyle y = -2$ (which I think is what you got).
  
  Then, the amplitude is the vertical distance from the sinusoidal axis to one of the peaks or troughs. So that's $\displaystyle 3$ (again, I think you got that).
  
  From these two facts, we can deduce that the equation of the graph will be of the form:
  $\displaystyle y=-2+3\sin$(something) or $\displaystyle y = -2+3\cos$(something)
  The question then refers to the 'starting point', based on a sine and then a cosine graph. I assume that means the point which corresponds to the point where $\displaystyle x = 0$ on the graphs of the basic graphs (the ones in my diagram).
  
  So, for sine, this is the point that corresponds to $\displaystyle (0,0)$ on the basic graph - in other words one of the points where the graph crosses the sinusoidal axis and is moving upwards in a direction from left to right. There's more than one such point, but the obvious one is $\displaystyle (30, -2)$. (Can you see that another one will be $\displaystyle (210,-2)$?)
  
  For the cosine function, the starting point will be the point corresponding to $\displaystyle (0,1)$ on the basic graph. So we're looking for a peak (maximum) value: the obvious one here is $\displaystyle (75,1)$.
  
  The period is the horizontal distance between two consecutive corresponding points on the graph. We said above that two possible starting points for the sine graph are $\displaystyle (30,-2)$ and $\displaystyle (210,-2)$. So the period is the distance between these two points. That's $\displaystyle 180^o$.
  
  Divide by $\displaystyle 360^o$ for the next answer: $\displaystyle \frac{180}{360}=\frac12$.
  
  If we divide $\displaystyle x$ by this fraction ($\displaystyle \tfrac12$) we get $\displaystyle 2x$, which gives us the coefficient of $\displaystyle x$ in the functions we're looking for. (A complete period for $\displaystyle y = \sin x$ is when $\displaystyle x$ goes from $\displaystyle 0^o$ to $\displaystyle 360^o$; so if we want a period of $\displaystyle 180^o$, we shall need values of $\displaystyle 2x$, because they will go from $\displaystyle 0^o$ to $\displaystyle 360^o$ when $\displaystyle x$ goes from $\displaystyle 0^o$ to $\displaystyle 180^o$.)
  
  The 'starting points' that we've worked out above tell us the angle that has to be added or subtracted from $\displaystyle x$ to give the correct starting value - the phase-shift.
  
  So, for the sine function we want to start at $\displaystyle x = 30$, so the function is
  $\displaystyle y=-2+3\sin(2[x-30])$
  And for the cosine function, we want to start at $\displaystyle x = 75$, so the function is
  $\displaystyle y=-2+3\cos(2[x-75])$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post where a user asks for help in determining the equation of a sinusoidal function from a given graph. The user specifically asks about finding the height (amplitude) of sine and cosine on the graph, the difference between sine and cosine, and the period of the function. Another user provides a detailed response explaining the differences between sine and cosine graphs, how to find the amplitude, the sinusoidal axis, the period, and the phase shift. The response also includes the equations for both the sine and cosine functions based on the given graph.
  
  Label: question-and-answer