## anonymous one year ago PLEASE HELP MULTIPLE QUESTIONS QUICKLY WILL FAN AND GIVE MEDAL Solve the equation on the interval 0≤θ<2π. sin(3θ)=−1

1. anonymous

@Directrix @Loser66 @zepdrix

2. anonymous

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

3. freckles

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

4. anonymous

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

5. freckles

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

6. freckles

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

7. anonymous

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

8. freckles

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

9. freckles

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

10. anonymous

at 3pi/2

11. freckles

yes

12. freckles

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

13. freckles

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

14. freckles

so replace x with 3 theta

15. freckles

and solve the linear equations

16. freckles

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

17. freckles

did you have any questions ?

18. anonymous

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

19. anonymous

sin* not sign

20. freckles

well you will have 3 solutions not just one

21. anonymous

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