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Finding the radian value of this angle which passes through a point

  1. Oct 29, 2013 #1
    1. The problem statement, all variables and given/known data
    The terminal arm of an angle in standard position passes through (-7,8). Find the radian value of the angle in the interval [0,2∏], to the nearest hundredth.


    2. Relevant equations
    sinθ = [itex]\frac{y}{r}[/itex]
    cosθ = [itex]\frac{x}{r}[/itex]
    tanθ = [itex]\frac{y}{x}[/itex]


    3. The attempt at a solution
    The terminal arm is in quadrant 2, and I found the side lengths to be -7,8, and [itex]\sqrt{113}[/itex] (hypotenuse). When I tried to find the value of θ I get different answers for different ratios.

    θ = sin[itex]^{-1}[/itex][itex]\frac{8}{\sqrt{113}}[/itex]
    = 0.85​

    θ = cos[itex]^{-1}[/itex][itex]\frac{-7}{\sqrt{113}}[/itex]
    = 2.29​

    θ = tan[itex]^{-1}[/itex][itex]\frac{8}{-7}[/itex]
    = -0.85​

    The correct one is θ=2.29. Why is this correct and not the others?
     
  2. jcsd
  3. Oct 29, 2013 #2
    The point -7,8 is in the second quadrant. So θ is less than π and greater than π/2.
     
  4. Oct 29, 2013 #3

    Mark44

    Staff: Mentor

    Because, as you pointed out, the angle is in the 2nd quadrant. This means that θ is between ##\pi/2## and ##\pi## (roughly, between 1.5 and 3).

    The sin-1 function returns an angle between ##-\pi/2## and ##\pi/2##, as does tan-1. The cos-1 function returns an angle between 0 ##\pi##, which was the correct interval for your angle.
     
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