Question

How do I solve this trigonometric equation?

Answer 1

\[\sqrt{3} + 2\cos 3\theta = 0\]

Answer 2

\[\Large\rm \sqrt{3} + 2\color{orangered}{(\cos 3\theta)} = 0\]Try to isolate your cosine 3theta first.
Undo the sqrt3 that's being added by subtracting, ya?

Answer 3

\[\Large\rm 2\color{orangered}{(\cos 3\theta)} = -\sqrt{3}\]What next? :d what do you think? How do we undo that multiplication of the 2?

Answer 4

\[\cos 3\theta = \frac{ \sqrt{3} }{ 2 }\]

Answer 5

Woops, your negative disappeared +_+\[\Large\rm \cos(3\theta)=-\frac{\sqrt3}{2}\]ya?

Answer 6

Think of it like this: \(\large\rm \cos(x)=-\frac{\sqrt3}{2}\)

What angle does this correspond to?
You have to think back to your unit circle, there are two places.

Answer 7

What's a unit circle? o_o As for the angle, it seems like it corresponds to maybe uhh....I think sixty degrees?

Answer 8

I mean 30. -.-

Answer 9

lol the unit circle?? :)
http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/2000px-Unit_circle_angles_color.svg.png

it's a circle of radius 1, we use it to reference all of our "special" angles.

Answer 10

bah my link didn't work >.<

Answer 11

Ok so 30 degrees.. but we want our cosine to be negative, so we need to be in quadrant 2 or 3.

Answer 12

Okay, so 150 degrees and 210?

Answer 13

Yes,
Did this problem give you a specific `interval` to work in?
Like 0 to 360 or anything?

Answer 14

My answer sheet says the degrees are 50, 70, 170, 190, 290, 310.

Answer 15

yeah, 0 < y < 360 degrees.

Answer 16

Here is what we've figured out so far....

Answer 17

For our angle, 3theta,\[\Large\rm 3\theta=150\]\[\Large\rm 3\theta=210\]But we're missing something here ^
We have to allow for full rotations.
So we write our solution as 150 plus any number of full spins. any multiple of 360 can be added on.\[\Large\rm 3\theta=150+360k\]\[\Large\rm 3\theta=210+360k\]

Answer 18

We're ultimately trying to find values for THETA, not 3theta.
So what's our next step? :)

Answer 19

Take a moment to soak all that in, ask question if you need :U I know trig is weird!

Answer 20

Okay I see where the 50 and 70 are from, but then I'm lost at the 360k part.

Answer 21

So we know that this is a solution.

Answer 22

But this is also a solution, ya?
Same location