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Math Help - Compound Angle Formulas

  1. #1
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    Compound Angle Formulas

    Can someone describe to me clearly, how you would solve these questions:

    1. Angles x and y are located in the first quadrant such that sinx=4/5 and cosy=7/25.
    a) determine an exact value for cos x
    b) determine an exact value for sin y
    c) determine an exact value for sin(x+y)

    2. angle x lies in the third quadrant, and tanx=7/24
    a) determine an exact value for cos2x
    b) determine an exact value for sin2x

    for 1 a and b I got it, but for 1 c I am not sure how to approach it
    Last edited by skeske1234; March 28th 2009 at 04:27 PM.
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  2. #2
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    Hello, skeske1234!

    1. Angles x and y are in the Quadrant 1 so that: \sin x=\tfrac{4}{5},\;\cos y=\tfrac{7}{25}

    a) Determine an exact value for \cos x\;\;{\color{blue}\tfrac{3}{5}}
    b) Determine an exact value for \sin y\;\;{\color{blue}\tfrac{24}{25}}
    c) Determine an exact value for \sin(x+y)

    We're expected to know this identity: . \sin(x+y) \:=\:\sin x\cos y + \cos x\sin y

    So we have: . \sin(x+y) \;=\;\tfrac{4}{5}\!\cdot\!\tfrac{7}{25} + \tfrac{3}{5}\!\cdot\!\tfrac{24}{25} \;=\;\tfrac{28}{125} + \tfrac{72}{125} \;=\;\tfrac{100}{125} \;=\;\frac{4}{5}



    2. Angle x is in the quadrant 3, and \tan x= \tfrac{7}{24}

    a) Determine an exact value for \cos2x
    b) Determine an exact value for \sin2x
    From \tan x = \tfrac{7}{24} in Quadrant 3, we have: . \begin{array}{c}\sin x \:=\:\text{-}\frac{7}{25} \\ \\[-3mm] \cos x \:=\:\text{-}\frac{24}{25}\end{array}


    a) We're expected to know that: . \cos2x \:=\:\cos^2\!x - \sin^2\!x

    So we have: . \cos2x \;=\;\left(\text{-}\tfrac{24}{25}\right)^2 - \left(\text{-}\tfrac{7}{25}\right)^2 \;=\;\tfrac{576}{625} - \tfrac{49}{625} \;=\;\frac{527}{625}


    b) We're expected to know that: . \sin2x \:=\:2\sin x\cos x

    So we have: . \sin2x \;=\;2\left(\text{-}\tfrac{7}{25}\right)\left(\text{-}\tfrac{24}{25}\right) \;=\;\frac{336}{625}

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  3. #3
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    FINALLY, a detailed answer.
    ok. thank you so much Soroban, really appreciate it!
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