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Math Help - 3 Hard Trigonometry Questions

  1. #1
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    3 Hard Trigonometry Questions

    Here are the questions (image below). Visual representations of the answers would be much appreciated!


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  2. #2
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    Hello, yeah!

    #2 took a lot of work . . . #3 is similar.
    Maybe someone else can find a more direct solution.


    2. The angle of elevation of the top of a tower is 38 from a point A south of it.
    The angle of elevation of the top of the tower is 29 from a point B east of it.
    Find the height of the tower if the distqance AB is 50m.
    Code:
                      P
                      o
                     *|  *
                    * |     *
                   *  |        *
                  *   |h          *
                 *    |              *
                *     |                 *
               *      |         x      29 *
              *       o   *   *   *   *   *   o B
             *38  *  Q              *
            *   * y         *  50
           * *     *
        A o
    The tower is: h = PQ.
    \angle PAQ = 38^o,\;\angle PBQ = 29^o
    \angle AQB = 90^o,\;AB = 50\text{m}
    Let:  x = QB,\;y = QA

    In right triangle PQA\!:\;\tan38^o \:=\:\frac{h}{y} \quad\Rightarrow\quad h \:=\:y\tan38^o .[1]

    In right triangle PQB\!:\;\;\tan29^o \:=\:\frac{h}{x} \quad\Rightarrow\quad h \:=\:x\tan29^o .[2]

    Equate [1] and [2]: . y\tan38 \:=\:x\tan29 \quad\Rightarrow\quad y \:=\:\frac{\tan29}{\tan38}\,x .[3]


    In right triangle AQB\!:\;\;x^2+y^2 \:=\:50^2

    Substitute [3]: . x^2 + \left(\frac{\tan29}{\tan38}\,x\right)^2 \:=\:2500 \quad\Rightarrow\quad x^2 + \frac{\tan^229}{\tan^238}\,x^2 \:=\:2500

    . . x^2\left(1 + \frac{\tan^229}{\tan^238}\right) \:=\:2500 \quad\Rightarrow\quad x^2\left(\frac{\tan^238 + \tan^229}{\tan^238}\right) \:=\:2500

    . . x^2 \:=\:\frac{2500\tan^238}{\tan^238 + \tan^229} \quad\Rightarrow\quad x\;=\;\frac{50\tan38}{\sqrt{\tan^238 + \tan^229}}


    Substitute into [2]: . h \;=\;\frac{\tan29}{\tan38}\cdot\frac{50\tan38}{\sq  rt{\tan^238 + \tan^229}}

    . . Grab your calculator . . .

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