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Thread: Compound Angle Formulas - Confused over evaluating

  1. #1
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    Compound Angle Formulas - Confused over evaluating

    I have done questions like these before, but the ones I did were only in quadrant I. I really wish I knew how to draw the triangles on here for you, so you see everything that I am working with.

    Question:
    If x is in the interval $\displaystyle (\frac{\pi}{2}, \pi) $ and y is in the interval $\displaystyle (\pi, \frac {3\pi}{2}) $ and cosx = $\displaystyle - \frac {5}{13} $ and tany = $\displaystyle \frac {4} {3} $, evaluate the following.

    *I will put the thing to evaluate, then the answer (according to the sheet) under it, then put my attempt after*

    a) sin (x + y)
    Answer: $\displaystyle - \frac{16}{65} $

    b) cos (x - y)
    Answer: $\displaystyle - \frac {33}{65} $

    c) tan (x - y)
    Answer: $\displaystyle \frac {56}{33} $



    Okay, now I will show my attempts (for some reason I got the answer for b, as the sheet answer for a. And the answer for b as the sheet answer for a.)


    a) $\displaystyle \sin (x + y) $
    = $\displaystyle \sin x \cos y + \cos x \sin y $
    = $\displaystyle ((\frac {12}{13}) (\frac {-4}{-5})) + ((\frac {-5}{13}) (\frac {-3}{-5}) $
    = $\displaystyle (\frac {48}{65}) + (\frac{-3}{13}) $
    = $\displaystyle \frac {33}{65} $


    b) $\displaystyle cos (x - y) $
    = $\displaystyle cosx cosy + sinx siny $
    = $\displaystyle ((\frac{-5}{13})(\frac{-4}{5}) + ((\frac {12}{13})(\frac{-3}{5})) $
    = $\displaystyle \frac {4}{13} + \frac {-36}{65} $
    = $\displaystyle \frac {-16}{65} $

    c) $\displaystyle tan (x - y) $
    = $\displaystyle \frac {tanx - tany} {1 + tanxtany} $
    = $\displaystyle \frac {(\frac {12}{13}) - \frac {-3}{-4}} {1 + (\frac {12}{13}) (\frac {-3}{-4}) } $
    = $\displaystyle \frac { \frac {9}{52} } {1+ \frac {9}{13} \frac {22}{13}} $
    = $\displaystyle \frac {9} {88} $
    Last edited by Xenophobe; Apr 19th 2009 at 03:16 PM.
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  2. #2
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    Hello, Xenophobe!

    If $\displaystyle x$ is in the interval $\displaystyle \left(\tfrac{\pi}{2}, \pi\right)$ and $\displaystyle y$ is in the interval $\displaystyle \left(\pi, \tfrac{3\pi}{2}\right)$

    and $\displaystyle \cos x = -\tfrac{5}{13}$ and $\displaystyle \tan y = \tfrac{4}{3} $, evaluate the following:

    $\displaystyle (a)\;\sin(x + y) \qquad (b)\;\cos(x - y) \qquad (c)\;\tan (x - y)$

    $\displaystyle x$ is in Quadrant 2, and $\displaystyle \cos x \:=\: -\frac{5}{13} \:=\:\frac{adj}{hyp}$
    We have: .$\displaystyle adj = -5,\;hyp = 13$
    . . Using Pythagorus: .$\displaystyle opp \:=\:12$


    $\displaystyle y$ is in Quadrant 3, and $\displaystyle \tan y \:=\:\frac{4}{3} \:=\:\frac{opp}{adj}$

    In Quadrant 3: .$\displaystyle opp = -4,\; adj = -3,\; hyp = 5$


    We have these values: . $\displaystyle \begin{array}{cccccccc}
    \sin x &=& \dfrac{12}{13} & & \sin y &=& \text{-}\dfrac{4}{5} \\ \\[-3mm]
    \cos x &=& \text{-}\dfrac{5}{13} & & \cos y &=& \text{-}\dfrac{3}{5} \\ \\[-3mm]
    \tan x &=& \text{-}\dfrac{12}{5} & & \tan y &=& \dfrac{4}{3} \end{array}$


    Try those in your compound angle formulas . . .

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