How can I solve a trigonometric equation using algebra?

[Peter G.]

Hi,

I was asked to solve for this:

sin (2x) = sin (0.5x)

For 0 to 270 degrees

I know how to solve it by plotting the two graphs, is there any other way of doing it, like, for example, algebraically?

Thanks,
Peter G.

 

[QuarkCharmer]

You could use the half angle and double angle identities for sin to expand both sides into something that you can work with.

For instance, on your left side:
sin(2x) is the same as sin(x+x) right?
So sin (x+x) = sinxcosx + sinxcosx, or 2sinxcosx

and you know how to solve a basic trig equation like that right?

 

[Peter G.]

I know the double angle rules but not the half angles, so I think I should've graphed them really. But I am having problems with two questions... I have only a vague idea on how to start: This is one of them:

The depth y meters of sea water in a bay at time t hours after midnight may be represented by the function:

y = a + bcos ((2π / k)t) where a and b are constants.

The water is at a maximum depth of 14.3 m at midnight and noon and is at a minimum depth of 10.3 m at 06:00 and at 18:00.

Write down the values of a, b and k:

I know a will shift the whole of the graph up or down, b will stretch it parallel to the y-axis and I think k will influence the period of the curve.

From then on however, I don't know what to do...

 

[John O' Meara]

For the first question, it is really $$sin\theta = sin\alpha \mbox{ The solution to these type of equations is } \theta = n180 + (-1)^n\alpha \mbox{ where n is an integer }$$. I hope this helps

 

[QuarkCharmer]

Well, since this is just the model of a simple harmonic, you can assume that the max at midnight and the min at 6 (and 18) are the max and mins, from that you should be able to get the amplitude of the function (b) and the constant a that shifts the graph up.

Inside the trig function you have ((2pi/k)t+0), let's call (2pi/k) B, and 0 C. 2pi over your "B" value must equal the period of the function, this can be used to determine the period of one wave if you pick a unit for time (hours perhaps). 2pi/2pi/k is really just k right? -C/B can be used to determine where your wave starts in relation to x=0.