Transformation of cosine graph

[Leah5467]

Hi! Please help!

We have to construct a formula for it,using a(cos(bx-c))+d The answer is -5(cos (pi/3)x). Can i assume that 6=angle in here? Is it possible to construct the formula if i assume that it shifts horizontally?Apologies for so many questions. I might make the same kind of mistakes if i am not clear about the concept ?
Thank you! Really a big thank you to the teachers who have helped me through the questions!

 

[Harry_the_cat]

Hi! Please help!
View attachment 11961
We have to construct a formula for it,using a(cos(bx-c))+d The answer is -5(cos (pi/3)x). Can i assume that 6=angle in here? Is it possible to construct the formula if i assume that it shifts horizontally?Apologies for so many questions. I might make the same kind of mistakes if i am not clear about the concept ?
Thank you! Really a big thank you to the teachers who have helped me through the questions!

You say "Can i assume that 6=angle in here?". Not sure what you mean by "angle" but the period is 6. So what is b?
What is the amplitude? So what is |a|?
Is there a vertical shift ? So what is d?

Now you can assume the original cos curve shifts to the right 3 units. Can you see why 3 units?

 

[Steven G]

Hi! Please help!
View attachment 11961
We have to construct a formula for it,using a(cos(bx-c))+d The answer is -5(cos (pi/3)x). Can i assume that 6=angle in here? Is it possible to construct the formula if i assume that it shifts horizontally?Apologies for so many questions. I might make the same kind of mistakes if i am not clear about the concept ?
Thank you! Really a big thank you to the teachers who have helped me through the questions!

When I first saw the 6 with the dots beneath it I thought that maybe the length of the dotted line was 6. But that is nonsense since the length of the dotted line is 5. So yes I would say that x was 6 and yes x represents the angles so the angle is 6.
But that wasn't enough for me. I try to back up my answers (else I write things like 1+4 =134) so I investigated further. I noted that the dotted line ended at a min values so I looked for one period of this curve. One period started at x=0 and ended at x equaling this questionable value of 6.
Since it started at x=0, I set (pi/3)x = 2pi. And when I solved it I got x=6. So most definitely that 6 is an x value.

And by the way ONE answer is y=-5(cos (pi/3)x). There are many answers for such a problem.

 

[Harry_the_cat]

Hi Jomo,

Just responding to your comment that "and yes x represents the angles so the angle is 6". In applications of sin/cos curves to concepts such as temperature, height of tides, height of a vertical spinning wheel etc, x does not represent angles at all. Usually it represents time or whatever he independent variable happens to be. Yes 6 is a point on the x-axis but not necessarily an angle. (I think that is clear when you look where the 5 and -5 are placed on the graph.) I suspect Leah was meaning "period" when she said "angle".

I do agree with the fact that there are many ways of writing a function which fits the curve. Leah suggested a horizontal shift which would lead to a different version of the function than the answer given, but would still be correct. I'm waiting to hear Leah's responses to my questions.

 

[Leah5467]

I think i meant "angle"?,as i was not sure if it could be other measurements. But now i see from your answer that it can be other things other than angle

 

[Steven G]

Hi Jomo,

Just responding to your comment that "and yes x represents the angles so the angle is 6". In applications of sin/cos curves to concepts such as temperature, height of tides, height of a vertical spinning wheel etc, x does not represent angles at all. Usually it represents time or whatever he independent variable happens to be. Yes 6 is a point on the x-axis but not necessarily an angle. (I think that is clear when you look where the 5 and -5 are placed on the graph.) I suspect Leah was meaning "period" when she said "angle".

I do agree with the fact that there are many ways of writing a function which fits the curve. Leah suggested a horizontal shift which would lead to a different version of the function than the answer given, but would still be correct. I'm waiting to hear Leah's responses to my questions.

I do not see your point-yet. I still think that you take sin and cosine of angles, not time. After all, in a calculator you set it to degrees or radians before you compute the sin or cos of something. So the something must be an angle(?). Please get back to me on this,

 

[Harry_the_cat]

The domain of the function f(x) = sin (x) is the set of real numbers. The range is [-1, 1].
In applications these x-values can be in any unit , they don't have to be measuring angles.

When the x-value does correspond to an angle, then there is a geometric interpretation. Note that sin(1) = sin (1 radian) not sin(1 degree).

I think the confusion or misunderstanding is that when we start learning trigonometry related to right-angled triangles the x values do represent angles (usually in degrees).

Sine functions can be used to model periodic functions such as tide heights, temperature, on a ferris wheel - ie things that go up and down periodically. These functions are often called circular functions and this is where the relationship with angles comes in. The x-values in these cases measure time though, not angles.

 

[Steven G]

The domain of the function f(x) = sin (x) is the set of real numbers. The range is [-1, 1].
In applications these x-values can be in any unit , they don't have to be measuring angles.

When the x-value does correspond to an angle, then there is a geometric interpretation. Note that sin(1) = sin (1 radian) not sin(1 degree).

I think the confusion or misunderstanding is that when we start learning trigonometry related to right-angled triangles the x values do represent angles (usually in degrees).

Sine functions can be used to model periodic functions such as tide heights, temperature, on a ferris wheel - ie things that go up and down periodically. These functions are often called circular functions and this is where the relationship with angles comes in. The x-values in these cases measure time though, not angles.

I need to think about what you are saying. It is usually the obvious that troubles me.

I do have one last question. In A = sin^-1(B). What are the As and Bs. I thought that A is always an angle? Is that too wrong??

 

[Harry_the_cat]

I need to think about what you are saying. It is usually the obvious that troubles me.

I do have one last question. In A = sin^-1(B). What are the As and Bs. I thought that A is always an angle? Is that too wrong??

It depends on how A and B were defined in the first place. In basic trigonometry, A would be an angle and B a real number in [-1, 1]

But take, for example, \(\displaystyle y = 2sin(\frac{\pi x}{6}) \)where y is the height (in metres) of the tide measured from a marker on a pylon of a bridge and x is the number of hours after midnight. Note that the period is 12 hours and the amplitude is 2 metres.

Now, for example, in finding y when x = 4 (ie find the height of water above the marker at 4am), you would need to use the inverse sine function. It will return the value of x and the unit will be hours after midnight. No angles involved.

The graph will have hours on the x-axis and metres on the y-axis.

 

[Harry_the_cat]

Ignore my second last paragraph above - a senior moment. What I meant to say is:

Now, for example, in finding x when y = 1.5 (ie find the time when the water is 1.5 metres above the marker), you would need to use the inverse sine function. It will return the value of x and the unit will be hours after midnight. No angles involved.