Solving Trig Equations

1. Oct 6, 2013

The question is "Find the real solutions θ for both of the following, a. sinθ=1/2, b. sin(7θ)=sin(5θ)"

On part a, I know that theta must be 30° or π/6 but I know that I need to mathematically show it and not sure how. And part b just has me head spinning, dont know where to start on that one.

Last edited by a moderator: Oct 7, 2013
2. Oct 6, 2013

haruspex

Anyway, for a., it's reasonable to assume π/6 is a solution; your problem is to find all the others. If you're not sure what the other solutions are, I suggest sketching a graph.
For b., I haven't tried it, but my approach would be to write 7 as 6+1 and 5 as 6-1, then use the usual trig formulae to expand the sines.

Last edited by a moderator: Oct 7, 2013
3. Oct 7, 2013

Staff: Mentor

sin(a)=sin(b) has two classes of solutions:
1) both points are in a region where the sine is increasing OR both are in a region where the sine is falling.
2) one point is in a region where the sine is increasing, the other is in a region where the sine is falling.
(solutions with sin(a)=sin(b)=1 or -1 are in both categories)

Those cases can be reduced to conditions for a-b and a+b respectively, and then you have a simple linear equation.

4. Oct 7, 2013

HallsofIvy

"On part a, I know that theta must be 30° or π/6 but I know that I need to mathematically show it and not sure how."

What, exactly to you feel would show that? Would putting "30 degrees" into a calculator and determining that "sin(30)" is equal to 0.5 be enough?

How about if you construct an equilateral triangle, so that all three angles are 60 degrees and all three sides have length 1. If you drop a perpendicular from one vertex to the opposite side, it divides the equilateral triangle into two right triangles with angles 30 and 60 degrees, hypotenuse of length 1 and leg opposite the 30 degree angle of length 1/2. The sine of 30 degrees is (1/2)/1= 1/2.

But I doubt that your teacher would require that. (It is not nearly so "nice" for other angles.) It should be enough to assert that "sin(60)= 1/2" or use a calculator to show that "arcsin(1/2)= 60 degrees".

But what I would consider the important part, you are not even considering! "sin(t)" is the y-component of the a point distance "t" around the unit circle. Drawing the horizontal line, y= 1/2, across the unit circle, it is easy to see that it crosses twice, once at "x" and the other at "180- x". So in addition to 60 degrees, another solution is 180- 60= 120. But sine is periodic with period 360 degrees to all solutions are of the form 60+ 360n or 120+ 360n degrees.

(You gave "60 degrees" first and then "$\pi/3$ radians". In fact, "degrees" are only used in problems that actually have angles measured in degrees. Most applications of "trig function" problems have little or no application to actual angles and the arguments are given in radians. I would consider the best answer to this problem $\pi/3+ 2\pi n$ and $2\pi/3+ 2\pi n$.)

5. Oct 7, 2013

Thank you for helping. :)

It is sine not cos mind you, and only +1/2 as a solution (i think). So that would be $\frac{\pi}{6} + 2 \pi n$ and $\frac{5\pi}{6} + 2 \pi n$ right?

6. Oct 8, 2013

haruspex

Fwiw, I have now tried what I suggested in post #2 for b and can report that it works extremely well.

7. Oct 8, 2013

Thanks, I gave what I think you were on about a go and have not got very far. Did you mean the general addition formumla ie for sin(A+B) etc?

I did...

$$sin(7\theta)=sin(5\theta) \\ sin(6\theta+1\theta)=sin(6\theta-1\theta) \\ sin(6\theta)cos(\theta)+sin(\theta)cos(6\theta)=sin(6\theta)cos(\theta)-sin(\theta)cos(6\theta) \\ sin(\theta)cos(6\theta)=-sin(\theta)cos(6\theta) \\ 2(sin(\theta)cos(6\theta))=0$$

I don't really know where I am going with that one :/

8. Oct 8, 2013

haruspex

If the product of two expressions is zero, what can you say about the two expressions?

9. Oct 8, 2013

That at least one of them must equal zero??

10. Oct 8, 2013

Staff: Mentor

Sure. What about the next step?

11. Oct 9, 2013

I am really not sure sorry. Maybe, if sinθ=0 then θ must equal 0 and/or π,. If cos(6θ)=0 then θ must equal (π/2)*6 = 3π and/or (3π/2)*6= 9π.

12. Oct 9, 2013

haruspex

It can't be 0 and pi, but yes, it could be 0 or pi or ....?
Again, it's or, not and, and there is an infinity of solutions.