Question

Help? I don't understand this at all...
Solve on the interval 0 ≤ θ < 2π
Cos(2θ-π/2)=-1

Answer 1

Basically you are finding values of theta where theta is in the interval 0 <= theta <= 2pi and satisfies the given equation. So basically what the equation says is cos(2theta - pi/2) = -1. So you find all values where cosine is -1 between 0 and 2pi (inclusive). Then these will be the values that 2theta - pi/2 must be equal to because these will be the values where cosine is -1 in the given interval. So you equation 2theta - pi/2 with whatever values you find where cosine is negative one in the given interval, and solve for theta.

Answer 2

@MoshiBunny
If you are completely lost, and don't know thing or don't understand a thing, then I can start you off =)

Answer 3

well you need to find the inverse or arc cos

then \[2\theta - \frac{\pi}{2} = \cos^{-1}(-1) \]

so

\[2 \theta - \frac{\pi}{2} = \pi\]

you should be able to solve for theta from here..

Answer 4

\[\cos \left( 2\theta - \frac{ \pi }{ 2 } \right)=-1 |0 \le \theta \le 2\pi \implies 2\theta - \frac{ \pi }{ 2 }=\pi\]Now solve for theta. Can you do that? @MoshiBunny

Answer 5

So I add pi/2 to both sides?

Answer 6

& divide by 2?

Answer 7

Yup.

Answer 8

What do you get?

Answer 9

thats correct...

Answer 10

so 2 theta = 3pi/2?

Answer 11

ya, now divide by 2.

Answer 12

theta = 3pi?
I feel like that's wrong..

Answer 13

It is wrong. It's 3pi/4

Answer 14

\[2\theta = \frac{ 3\pi }{ 2 } \rightarrow \theta = \frac{ 3\pi }{ 2 } \div 2 = \frac{ 3\pi }{ 2 } \times \frac{ 1 }{ 2 }=\frac{ 3\pi }{ 4 }\]You probably made a division mistake. @MoshiBunny

Answer 15

Yeah, I forgot I had to do the reciprocal ..

Answer 16

Ya. So you get how the problem was solved? You are basically looking for where the cosine graph is -1 in the interval [0, 2pi]. Graphing it will show that only at pi cosine is -1. That means that the 2theta - pi/2 inside the brackets must equal to pi. So you equate it as 2theta - pi/2 = pi and solve for theta.

You get what was done here to get the answer? @MoshiBunny

Answer 17

So, it would be theta=3pi/4+2kpi?

Answer 18

No not k. Because we are given an interval to solve for theta for and that interval is [0,2pi], then we only suggest those values of theta in that interval which satisfy the equation. If we weren't told to find theta in a certain interval, only then would we include 'k'.

@MoshiBunny

Answer 19

So it would be just theta = 3pi/4

Answer 20

Oh.. so there aren't any, like "extra answers"?

Answer 21

Exactly. The answer with 'k' in it is a general solution, meaning it gives solutions over a domain. Since we are given an interval, we only solve for values satisfying the equation in that interval. So IN THIS CASE, there is no extra answers, just one.

Answer 22

Thank you! I finally get this :D

Answer 23

Fan me? =D

Answer 24

Sure :]

Answer 25

Btw, if the question has it equal to 0 instead of -1, in the interval [0, 2pi], how many values of theta would we get?

Answer 26

@MoshiBunny

Answer 27

1? .. uh.. hehe....

Answer 28

Nope. We would get 2 answers for theta. This is because in the interval [0, 2pi], the graph of cosine is 0 at two different points, at pi/2 and 3pi/2. So then we solve the inner bracket of 2theta - pi/2 = pi/2 and 2theta - pi/2 = 3pi/2 and that will give us 2 different answers for theta which satisfy the equation in the interval [0,2pi]. Get it?

You should really look at the graph of cosine and sine to get a good understanding of this.
@MoshiBunny

Answer 29

Oh.. I looked at my unit circle and I didn't see 3pi/2 on it so I said 1 answer. :O