Thread: Function y=3Cos2x solve for x when 3Cos2x=2.0

1. Function y=3Cos2x solve for x when 3Cos2x=2.0

Hello, I have a function y=3Cos2x
Now I have a graph which I plotted, it makes a nice curve.

Here are the x Values: 0, πc/10, πc/5, (3πc)/10, (2πc)/5, πc/2, (3πc)/5, (7πc)/10, (4πc)/5, (9πc)/10, and πc

The corresponding Y values after running through the function in order from the first x value are: 3.00, 2.43, 0.93, -0.93, -2.43, -3.00, -2.43, -0.93, 0.93, 2.43, 3.00

Now the question asked for the solutions of : 3Cos2x=2.0

My first thought was to look at my graph and see where the line y=2.0 intercepts the curve and found the corresponding x coordinates, I did and obtained,

0.13πc, and 0.86πc, which when you multiply by 180 to turn into degrees, I get 23.4 degrees, and 154.8 degrees, which I'm fairly sure is correct
as my book has x = 24 degrees, and x = 156 degrees, so I'm guessing that my graph was just a bit to the left and it's human error.

However, they also have x = (2πc)/15 and x = (13πc)/15

Am I missing something? how did they get that? Did I have to use the function and solve it?

The scale on my graph is πc/10 for 2 cm on the X axis and 2cm for 1 unit on the Y axis.

2. Originally Posted by richiesmasher
Hello, I have a function y=3Cos2x
Now I have a graph which I plotted, it makes a nice curve.

Here are the x Values: 0, πc/10, πc/5, (3πc)/10, (2πc)/5, πc/2, (3πc)/5, (7πc)/10, (4πc)/5, (9πc)/10, and πc

The corresponding Y values after running through the function in order from the first x value are: 3.00, 2.43, 0.93, -0.93, -2.43, -3.00, -2.43, -0.93, 0.93, 2.43, 3.00

Now the question asked for the solutions of : 3Cos2x=2.0

My first thought was to look at my graph and see where the line y=2.0 intercepts the curve and found the corresponding x coordinates, I did and obtained,

0.13πc, and 0.86πc, which when you multiply by 180 to turn into degrees, I get 23.4 degrees, and 154.8 degrees, which I'm fairly sure is correct
as my book has x = 24 degrees, and x = 156 degrees, so I'm guessing that my graph was just a bit to the left and it's human error.

However, they also have x = (2πc)/15 and x = (13πc)/15

Am I missing something? how did they get that? Did I have to use the function and solve it?

The scale on my graph is πc/10 for 2 cm on the X axis and 2cm for 1 unit on the Y axis.
Please tell us the actual wording of the exercise. In particular, were you told to solve by looking at a graph? I would expect you to use the inverse cosine function; have you learned about that yet? And how did you get the numbers for the graph?

What does your superscript c, as in "πc/10", mean?? I suppose it must mean "radians", but normally when we use radians, that is just assumed, and no symbol is needed. Were you taught to use this symbol, or did you make it up?

At any rate, if the answer is supposed to be in degrees, and you are expected to find an approximate value by manual graphing, I would do everything in degrees rather than radians. That will also result in numbers that are easier to work with.

It appears that they used degrees, and then converted their rounded degree values to radians; that accounts for their (2πc)/15 and (13πc)/15, which are equal to your decimal values.

So your work seems to be entirely correct, just starting with radians where they started with degrees. That is enough to explain the slight difference in numbers. (The "human error" involved is the reason we don't normally solve by graphing. That method is appropriate mostly as a way to become familiar with the graphs.)

3. Originally Posted by Dr.Peterson
Please tell us the actual wording of the exercise. In particular, were you told to solve by looking at a graph? I would expect you to use the inverse cosine function; have you learned about that yet? And how did you get the numbers for the graph?

What does your superscript c, as in "πc/10", mean?? I suppose it must mean "radians", but normally when we use radians, that is just assumed, and no symbol is needed. Were you taught to use this symbol, or did you make it up?

At any rate, if the answer is supposed to be in degrees, and you are expected to find an approximate value by manual graphing, I would do everything in degrees rather than radians. That will also result in numbers that are easier to work with.

It appears that they used degrees, and then converted their rounded degree values to radians; that accounts for their (2πc)/15 and (13πc)/15, which are equal to your decimal values.

So your work seems to be entirely correct, just starting with radians where they started with degrees. That is enough to explain the slight difference in numbers. (The "human error" involved is the reason we don't normally solve by graphing. That method is appropriate mostly as a way to become familiar with the graphs.)
No I wasn't told to look at the graph, It just said solve.

But I figured I could look at the graph to see where those two values were... maybe that's a shortcut and I should know the other method.

Yes I do know the inverse cosine function, But I don't know how to solve the equation using it, So like I said I used the graph.

Yes that symbol means Pi radians over 10.

4. Originally Posted by richiesmasher
No I wasn't told to look at the graph, It just said solve.

But I figured I could look at the graph to see where those two values were... maybe that's a shortcut and I should know the other method.
Actually, the problem as you show it does clearly tell you to solve "using your graph". That's what I wanted to know; they did tell you what method to use. And what you did was exactly what you were told, even labeling the graph in radians rather than degrees. You've done nothing wrong.

You haven't told me how you actually drew the graph, but presumably you used either a calculator or a table to find values given angles in radians, plotted them, and drew a smooth curve. Such a hand-drawn graph will never be perfect, so you expect only approximate answers.

The curious thing is that, as I said before, they appear to have done their work using degrees first, as I would, rather than the way they told you to. Since you did what they asked, and got reasonably close answers, there is nothing to be concerned about.

Originally Posted by richiesmasher
Yes I do know the inverse cosine function.

Yes that symbol means Pi radians over 10.
But they don't use that symbol in the book, right? Don't use it; it's just confusing. (Unless your teacher wants you to, for some reason. But I've never seen it before.)

When they want you to use the inverse cosine, they will tell you. But I suppose there's nothing wrong with trying it, to check your answer.

5. Originally Posted by Dr.Peterson
Actually, the problem as you show it does clearly tell you to solve "using your graph". That's what I wanted to know; they did tell you what method to use. And what you did was exactly what you were told, even labeling the graph in radians rather than degrees. You've done nothing wrong.

You haven't told me how you actually drew the graph, but presumably you used either a calculator or a table to find values given angles in radians, plotted them, and drew a smooth curve. Such a hand-drawn graph will never be perfect, so you expect only approximate answers.

The curious thing is that, as I said before, they appear to have done their work using degrees first, as I would, rather than the way they told you to. Since you did what they asked, and got reasonably close answers, there is nothing to be concerned about.

But they don't use that symbol in the book, right? Don't use it; it's just confusing. (Unless your teacher wants you to, for some reason. But I've never seen it before.)

When they want you to use the inverse cosine, they will tell you. But I suppose there's nothing wrong with trying it, to check your answer.
How do I use the cosine inverse to check my answer in this case?

6. Originally Posted by richiesmasher
How do I use the cosine inverse to check my answer in this case?
Give it a try! I imagine they are leading up to that, and haven't yet given any examples, but you should find some later in the book.

Just do the same thing you usually do to solve an equation: undo one operation at a time. You are solving 3Cos2x=2; so first divide by 3, then undo the cosine by taking the inverse cosine of each side (but think about what other values the angle can have to give the same cosine); then divide by 2. That part about other values is the hard part, which they will spend some time on. But you should get at least the first value easily.

7. Originally Posted by Dr.Peterson
Give it a try! I imagine they are leading up to that, and haven't yet given any examples, but you should find some later in the book.

Just do the same thing you usually do to solve an equation: undo one operation at a time. You are solving 3Cos2x=2; so first divide by 3, then undo the cosine by taking the inverse cosine of each side (but think about what other values the angle can have to give the same cosine); then divide by 2. That part about other values is the hard part, which they will spend some time on. But you should get at least the first value easily.
Indeed I did get the first value easily 24.1, ok now I will work on the hard part.....I imagine it has something to do with the graph of the cos function... I know it has to do with this concept,
Like in the sine graph, sine 90 is 1, sine -90 is -1

8. Originally Posted by richiesmasher
Indeed I did get the first value easily 24.1, ok now I will work on the hard part.....I imagine it has something to do with the graph of the cos function... I know it has to do with this concept,
Like in the sine graph, sine 90 is 1, sine -90 is -1
Yes, you can relate it to the graph, and in particular to symmetry. There are several ways to think about it; the one you figure out for yourself will probably be the best way for you, so I'll let you have the fun of discovering it. Let me know!

9. Originally Posted by Dr.Peterson
Yes, you can relate it to the graph, and in particular to symmetry. There are several ways to think about it; the one you figure out for yourself will probably be the best way for you, so I'll let you have the fun of discovering it. Let me know!
I did figure out an angle, using the circle and reference triangles, it would be equivalent to 270 +24.1 degrees which is 335.9 degrees and the cosine of that indeed is the same as the cosine of 24.1, however it's not close to the answer in the book which is 156 degrees.

10. Originally Posted by richiesmasher
I did figure out an angle, using the circle and reference triangles, it would be equivalent to 270 +24.1 degrees which is 335.9 degrees and the cosine of that indeed is the same as the cosine of 24.1, however it's not close to the answer in the book which is 156 degrees.
At the point where you applied the inverse cosine, you found that 2x = 48.2°. This is where you have to find the other angle, not after solving for x. You are not working with the cosine of x itself.

The two angles that have the same cosine are 48.2° and -48.2°. But any angle coterminal with these will also have the same cosine. You need to find one so that x is in the desired interval (0° to 180°).

Why did you think adding 270 would work? Reference angles are always related to 0 or 180. You may have looked at something incorrectly.

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