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MathCurious

  • 2 years ago

sin (0.77x) = -1/2 How did they get 0.77x = 3.665 and 5.760? Someone please help show me the steps of their work so I can see it.

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  1. theEric
    • 2 years ago
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    I think there are two ways. The first is memorizing what you take the sine of to get -1/2. The other ways is using the inverse sine function. Because\[sin(77x)=-\frac{1}{2}\], you can say \[77x = sin^{-1}(-\frac{1}{2})\]

  2. theEric
    • 2 years ago
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    This only helps if you have a calculator..

  3. MathCurious
    • 2 years ago
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    Hello, thank you for answering. This is what I initially thought as well. however, the inverse sine of negative one half gave me: -0.5235987756 radians. Above it shows different answers for the solution.

  4. MathCurious
    • 2 years ago
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    So I am baffled how my text came up with such a number.

  5. theEric
    • 2 years ago
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    Huh! I'm foggy on trigonometry, sorry!

  6. MathCurious
    • 2 years ago
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    Thanks anyways.

  7. theEric
    • 2 years ago
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    I used wolfram alpha to solve for \[a\]in\[sin(a)=-\frac{1}{2}\] and it produced \[x=\frac{7\pi}{6}+2\pi n_1\]and\[x=\frac{11\pi}{6}+2\pi n_2\]

  8. theEric
    • 2 years ago
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    The first is 3.6651914291880921115397506138260866982300309659376234...

  9. theEric
    • 2 years ago
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    when \[n_1=0}\]

  10. MathCurious
    • 2 years ago
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    Oh I see.. The 3.665 and 5.760 are just decimal equivalents of the fraction. Thanks for shedding some light on it.

  11. theEric
    • 2 years ago
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    when \[n_1=0\] I mean

  12. theEric
    • 2 years ago
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    If you got it, then congratulations! I'm still a bit confused! :)

  13. theEric
    • 2 years ago
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    The second resolves to 5.7595865315812876038481795360124219543614772321876940... when \[n_2=0\]

  14. MathCurious
    • 2 years ago
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    Haha. I got it. It is saying. The sine function is equal to -1/2 in two different quadrants. In a circle (degrees) it is located at 7pi/6 and 11pi/6 These are equivalent to a 30 degree angle (pi/6) in the first quadrant. 7pi/6 = 3.665

  15. MathCurious
    • 2 years ago
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    Excuse me, those are radians!

  16. MathCurious
    • 2 years ago
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    Well thanks for guiding me. I got it. Medaled.

  17. theEric
    • 2 years ago
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    Cool! Would you just have to memorize \[\frac{7\pi}{6}\] and \[\frac{11\pi}{6}\]? I think you would have to...

  18. theEric
    • 2 years ago
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    Ah well, I'll recap another day! I'm glad I could be of some service! Take care!

  19. MathCurious
    • 2 years ago
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    I wouldn't really memorize, but it's not a bad idea if you have that capacity! :) If anything it would be convenient. However, there is a method to finding it with logic. If the Y-coordinate is -1/2 then the x must be \[\sqrt{3}\div2\] this is assuming we understand the basic triangles. This information tells us with an x-coord of 1/2 and a y-coord of \[\sqrt{3}\div2\] it is a 30 degree angle (in quadrant I). However the problem needs a solution for where Y is negative. |dw:1350355894146:dw| It is true in two quadrants. The formula for DEGREES to RADIANS is:( DEGREE*PI )/ 180 since quadrant III begins from 180 and ends at 270 AND we knew it is essentially a 30 degree angle then we can add 30 DEGREES to 180. This becomes a 210 angle. Divide 270 by 180 and you get 7/6. Don't forget to multiply it by Pi. Hopefully that shows how to get an angle in another quadrant. There are of course other tricks to get there. |dw:1350356151031:dw| You could see this visually. You could just directly reflect the angle into an opposite quadrant by making a straight line out of it. Here it has 180 degrees added to it. So you don't need to just simply memorize radians.

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