Combining sinusoidal functions

[mdem1234]

Hi

I hope this question belongs here and not under the 'calculus' topic.. I wasnt sure because I am studying it in calculus..

I just need help with combining sinusoidal functions.

The question is

For each of the following functions, find the mean value, period and amplitude and hence
sketch the graph.

(a) 2 + 4 sin (4¼(x ? 1)).
(b) 3 + 5 sin (2¼(x ? 1)).
(c) 2 sin 3x ? 4 cos 3x ? 5.
(d) sin 2x ? 2 cos 2x + 6.

I can do part (a) and (b) easily, because they are already in the right form for me to easily derive the mean, period etc from. But parts (c) and (d) I'm having a lot of trouble with. I know that I first need to combine the sin and cos function so that they become 1 function, and then i can find the mean, period etc and then sketch the graph.

Any help would be great!

Thanks!

 

[DrMike]

If you have, say 7 sin(4x) + 24 cos(4x), then this must be A sin(4x+B). You need to find A and B. (Or, you could write it as C cos(4x+D) )

Using the formula for sin(x+y), I get

A sin(4x+B) = A sin(4x) cos(B) + A cos(4x) sin(B).

Now I have two equations :

A cos(B) = 7
A sin(B) = 24.

Do you know what to do from here?

 

[mdem1234]

Reading over my notes it looks like i would first take 7^2 + 24^2 and then square root the sum to get 25, which is what A is equal to?

That should give cos (B) = 7/25 and sin (B) = 24/25

Is this right? Im kinda just guessing from my notes. Would you be able to explain why we do this (if im right) and where do i go from here?

Thanks

 

[DrMike]

Reading over my notes it looks like i would first take 7^2 + 24^2 and then square root the sum to get 25, which is what A is equal to?

That should give cos (B) = 7/25 and sin (B) = 24/25

Is this right? Im kinda just guessing from my notes. Would you be able to explain why we do this (if im right) and where do i go from here?

Thanks

That's right!

The reason is this :

In almost any system of two equations with two variables, the quickest way to solve it is to rearrange/combine things so one of the variables disappears.

Here, you have

A cos(B) = 7
A sin(B) = 24

If I square these, I get

A[sup:3lt24u95]2[/sup:3lt24u95] cos[sup:3lt24u95]2[/sup:3lt24u95](B) = 7[sup:3lt24u95]2[/sup:3lt24u95]
A[sup:3lt24u95]2[/sup:3lt24u95] sin[sup:3lt24u95]2[/sup:3lt24u95](B) = 24[sup:3lt24u95]2[/sup:3lt24u95].

Adding these, I get

A[sup:3lt24u95]2[/sup:3lt24u95](cos[sup:3lt24u95]2[/sup:3lt24u95](B)+sin[sup:3lt24u95]2[/sup:3lt24u95](B))=7[sup:3lt24u95]2[/sup:3lt24u95]+24[sup:3lt24u95]2[/sup:3lt24u95]

and hey presto, the B disappears!

Then A[sup:3lt24u95]2[/sup:3lt24u95]=625, so A=25 (it doesn't matter for this problem which of A=-25 and A=25 I choose. If you are curious about this, try taking A=-25 and see what happens)

Then, substituting A=25 back into the original equations, you get

25 cos(B) = 7
25 sin(B) = 24

as you noted.