# Trigonometric Equations

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.

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?

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...

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

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.