Use a graphing calculator to solve the equation -3 cos t = 1 in the interval from

- anonymous

Use a graphing calculator to solve the equation -3 cos t = 1 in the interval from

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- anonymous

\[0\le \theta \le 2x\]

- anonymous

Round to the nearest hundredth and show your work.

- freckles

is x suppose to be pi?

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## More answers

- anonymous

oh yeah sorry it is

- freckles

\[\cos(t)=\frac{-1}{3} \text{ on } 0 \le t \le 2\pi \\ \text{ one solution can be found just by taking } \arccos( ) \text{ of both sides } \\ \text{ other one we can find \it by using that } \\ \text{ cosine is even and has period } 2 \pi \\ \cos(-t)=\cos(t) \\ \text{ so we have } \\ \cos(-t+2\pi)=\frac{-1}{3} \\ \text{ take } \arccos( ) \text{ of both sides and solve for } t \]

- anonymous

@freckles can you keep going cuz i really dont know how to do this

- anonymous

I fanned you and gave you a medal

- freckles

well were you not able to the first solution?
it is just taking arccos( ) of both sides

- freckles

\[\cos(t)=\frac{-1}{3} \text{ we could say one solution is } t=\arccos(\frac{-1}{3})\]

- anonymous

No im really confused, that's why i like when people go step by step

- freckles

Oh I thought I gave you step by step

- freckles

all I did there was take arccos( ) of both sides for that one solution

- freckles

just as I said to do in the steps above

- anonymous

you did, but is that the answer?

- freckles

that is one solution
i also said how to find the other

- freckles

use the fact that cos is even and has period 2pi

- anonymous

so the answer is -1/3?

- freckles

how did you get that?

- anonymous

No i'm asking you, from your step by step that's what is above that everything is equalling -1/3

- freckles

the equation
is cos(t)=-1/3
we aren't solving for cos(t)
if we were then we would done and the answer is -1/3
we are solving for t

- freckles

one solution is given by taking arccos( ) of both sides of the equation cos(t)=-1/3

- anonymous

ok how do we do that?

- freckles

one solution is given by t=arccos(-1/3)

- freckles

and then the other solution is given by using the fact that cos is even and has period 2pi

- anonymous

t=4.37255207Â±2Ï€n

- freckles

let me give you an easy example to follow:
\[\cos(\theta)=\frac{-1}{2} , \text{ where } 0 \le \theta \le 2 \pi \\ \text{ we know we are suppose to get solutions } \theta=\frac{2 \pi }{3} ,\frac{4\pi}{3} \\ \text{ we see we can get these solutions by doing : } \\ \text{ Answer one given by just takinng } \arccos( ) \text{ of both sides } \\ \theta=\arccos(\frac{-1}{2}) \\ \\ \text{ now the other solution comes from using the fact that } \\ \text{ cosine is even function and has period } 2\pi \\ \cos(-\theta+2\pi)=\frac{-1}{2} \\ \text{ solve for } \theta \text{ by first taking } \arccos( ) \text{ of both sides } \\ -\theta+2\pi=\arccos(\frac{-1}{2}) \\ \text{ now subtract } 2\pi \text{ on both sides } \\ -\theta=\arccos(\frac{-1}{2})-2\pi \\ \text{ last step multiply -1 on both sides } \\ \theta=-\arccos(\frac{-1}{2})+2\pi \\ \\ \text{ so the solutions to } \cos(\theta)=\frac{-1}{2} \text{ on } 0 \le \theta \le 2\pi \text{ are } \\ \theta=\arccos(\frac{-1}{2}) \text{ or } \theta=-\arccos(\frac{-1}{2})+2\pi \\ \text{ we can simplify these solutions } \\ \text{ We know } \arccos(\frac{-1}{2})=\frac{2\pi}{3} \\ \\\ \text{ so we have } \theta=\frac{2\pi}{3} \text{ or } \theta=-\frac{2\pi}{3}+2\pi=\frac{4\pi}{3} \\ \text{ just as we got way above just from using the unit circle }\]

- freckles

|dw:1437253274947:dw|

- freckles

anyways you can use the exact same process I used above

- freckles

just take arccos() of both sides to get one solution

- freckles

then rewrite the equation as cos(-t+2pi)=whatever
where whatever is between -1 and 1 (inclusive)
and take arccos( ) of both sides and then solve that resulting equation for t

- freckles

\[\cos(t)=\frac{-1}{3} \\ \text{ one solution is } t=\arccos(\frac{-1}{3} ) \\ \text{ the other solution comes from solving } \cos(-t+2\pi)=\frac{-1}{3}\]

- freckles

if I take arccos( ) of both sides of your second equation
-t+2pi=arccos(-1/3)
can you try to solve this for t?

- freckles

it is just two steps

- anonymous

Is the answer theta 4/2pi?

- freckles

do you know how to solve
-x+4=6 for x?

- anonymous

yes

- freckles

like you would subtract 4 on both sides
-x=6-4
-x=2
and last step is multiply -1 on both sides
x=-2
like this is the same type of equation here:
\[-t+2\pi=\arccos(\frac{-1}{3} ) \\ \text{ subtract } 2\pi \text{ on both sides } \\ -t =\arccos(\frac{-1}{3})-2\pi \\ \text{ multiply -1 on both sides } \\ t=-\arccos(\frac{-1}{3})+2\pi\]

- freckles

you have both solutions
you can use your calculator to approximate them if you want

- freckles

I think the instructions above do say to round to nearest hundredth so you will have to break out the calculator

- anonymous

Will you help me to the final answer?

- freckles

I gave you both exact solutions...
\[t=\arccos(\frac{-1}{3}) \\ t=-\arccos(\frac{-1}{3})+2\pi\]
?

- freckles

oh are you saying you don't know how to plug it in the calculator?

- freckles

there should be arcos typed on your calculator right above the cos button

- freckles

arccos*

- freckles

are it could be written as cos^(-1)

- freckles

do you see it?

- freckles

to access it on some calculators you have to press (shift or 2nd)
can't remember which one

- anonymous

No i cant get a good answer it keeps saying error

- anonymous

t=8.19381854

- freckles

what calculator do you use
i can try to look it up and tell you what buttons to push

- anonymous

t=8.19381854

- anonymous

is that right?

- freckles

oh i was waiting on you to tell me the calculator type but no

- anonymous

It's on a website cuz i dont have a calculator

- freckles

arccos(-1/3)=1.91
by using a calculator
so we already have that as a solution
now replace the arccos(-1/3) in the other solution with 1.91
\[-1.91+2\pi\]
you can do 3.14 as an approximation for pi
3.14*2=6.28
so what is -1.91+6.28

- anonymous

4.37

- freckles

yes so remember you have two answers

- anonymous

So 4.37 is one, what's the other?

- freckles

:(

- anonymous

what?

- freckles

I just feel like you haven't listened to me
\[t=\arccos(\frac{-1}{3}) \approx 1.91 \\ t=-\arccos(\frac{-1}{3})+2\pi \approx -1.91+2(3.14) =4.37\]

- anonymous

No I have, im just learning. Im sorry

- freckles

ok well I hope you understood some of what I said

- freckles

I have to leave now
any questions before I go

- anonymous

Im gonna overview it right now, thank you very much for your help i appreciate it!

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