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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

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  1. anonymous
    • one year ago
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    @Directrix @Loser66 @zepdrix

  2. anonymous
    • one year ago
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    Take the inverse sin of both sides and then solve for θ.

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

  4. anonymous
    • one year ago
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    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
    • one year ago
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    do you know how to solve sin(x)=-1 on 0 <=x<6pi?

  6. freckles
    • one year ago
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    if not do you know how to solve sin(x)=-1 on 0<=x<2pi?

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

  9. freckles
    • one year ago
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    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
    • one year ago
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    at 3pi/2

  11. freckles
    • one year ago
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    yes

  12. freckles
    • one year ago
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    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
    • one year ago
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    and what we actually want to solve for is theta not x

  14. freckles
    • one year ago
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    so replace x with 3 theta

  15. freckles
    • one year ago
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    and solve the linear equations

  16. freckles
    • one year ago
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    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
    • one year ago
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    did you have any questions ?

  18. anonymous
    • one year ago
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    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
    • one year ago
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    sin* not sign

  20. freckles
    • one year ago
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    well you will have 3 solutions not just one

  21. anonymous
    • one year ago
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    Ah yes, but that can be done by cumulatively adding 2pi before dividing by 3. One extra step.

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