Question

PLEASE HELP MULTIPLE QUESTIONS QUICKLY WILL FAN AND GIVE MEDAL

Solve the equation on the interval 0≤θ<2π.
sin(3θ)=−1

Answer 1

@Directrix @Loser66 @zepdrix

Answer 2

Take the inverse sin of both sides and then solve for θ.

Answer 3

\[\text{ Let } x=3 \theta \text{ so } \theta=\frac{x}{3} \\ \text{ we want to solve } \sin(x)=-1 \text{ on the interval } 0 \le \frac{x}{3} <2 \pi\]

Answer 4

Guys I've been sick so I missed a lot of school and this homework is due at 12 and I'm really struggling so please explain it so I can understand it quickly

Answer 5

do you know how to solve sin(x)=-1 on 0 <=x<6pi?

Answer 6

if not do you know how to solve sin(x)=-1 on 0<=x<2pi?

Answer 7

no ive been absent for a while I'm trying to see if someone can work it out step by step so I can understand whats going on

Answer 8

https://www.mathsisfun.com/geometry/images/circle-unit-radians.gif
can you tell me when the y coordinate is -1 here?

Answer 9

For example
If I asked you to look at that chart and tell you to tell me when the y-coordinate is 1, the answer I would be looking for is pi/2. Which means sin(pi/2)=1

Answer 10

at 3pi/2

Answer 11

yes

Answer 12

sin(x)=-1 when x=3pi/2 if we were only looking at 0<=x<2pi
but we also need to find the roots in 2pi<=x<4pi
and the one in 4pi<=x<6pi
so that means there are two more roots to be found
guess what we can just find these by adding 2pi to our already found root
and then 2pi again
x=3pi/2
x=3pi/2+2pi
x=3pi/2+2pi+2pi or 3pi/2+4pi

now we are not done
remember we let x=3 theta

Answer 13

and what we actually want to solve for is theta not x

Answer 14

so replace x with 3 theta

Answer 15

and solve the linear equations

Answer 16

solve all three linear equations below:
\[3 \theta=\frac{3 \pi}{2} \\ 3 \theta=\frac{3\pi}{2}+2\pi \\ 3 \theta=\frac{3\pi}{2}+4\pi\]

Answer 17

did you have any questions ?

Answer 18

I think I might have a simpler solution. It also requires less steps.
First, let's get rid of the 3θ and substitute it with x. Easy enough, right?
\[\sin(x)=-1\]
Now let's think of the unit circle here. What angle gives us -1 when we take the sign of it? 3pi/2 right? Perfect!
\[x=\frac{ 3\pi }{ 2 }\]
We can check this by taking the sign of it.\[\sin(x)=\sin(\frac{ 3\pi }{ 2 })=-1\]
Look at that.

Now that we have x, substitue that 3θ back in there.
\[3\theta=\frac{ 3\pi }{ 2 }\]

Therefore\[\theta=\frac{ 3\pi }{ 6 } \in [0, 2\pi)\]

Answer 19

sin* not sign

Answer 20

well you will have 3 solutions not just one

Answer 21

Ah yes, but that can be done by cumulatively adding 2pi before dividing by 3. One extra step.