Question

cos(3x) = -1
Part 1: using an appropriate inverse trigonometric expression, write an equation that defines the value of 3x.

Part 2: solve this equation to find all possible values of the angle 3x.

Part 3: use algebra to find all values of x between 0 and 2pi that satisfy this equation.

Answer 1

Remember inverse functions back from algebra? :)
To find the inverse we would `swap` the variables, and then solve for our new y.

Same idea here, we change our cosine to inverse cosine, and swamp the argument with the other side.

\[\large \cos(3x)=-1 \qquad\to\qquad \arccos(-1)=3x\]

Answer 2

swap* not swamp lol

Answer 3

That makes a lot of sense, Thank you! :D Could you help me with the other questions too?

Answer 4

Hmm part 2, let's seeee.

I actually think it's easier to leave it in cosine form to solve the second part.

We have \(\large \cos(3x)=-1\)
Where 3x is just some angle, let's call it theta for right now,\[\large \cos \theta=-1\]
Do you remember your special angles?
What angle when you take the cosine of it, gives us -1?

Answer 5

Oh actually, that's part 3.. my bad.
For part 2, they just want you to solve for x.

So in this case, we would just divide both sides by 3 right?

\[\large x=\frac{\arccos(-1)}{3}\]

Answer 6

so x= pi/3 ?

Answer 7

Yesss very good! :)

Answer 8

Thank you! So for part 3, is the special angle pi?

Answer 9

Yes it is. If we had done it that way, we would have determined that \(\large \theta=\pi\).

Or in other words, \(\large 3x=\pi\).
And then dividing by 3 would get us the same answer.

But it looks like you already figured that out based on the other method we used from here\[\large x=\frac{\arccos(-1)}{3}\]

So it's all good :D

Answer 10

Okay! Thanks so much! I totally understand what to do now! :D