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Math Help - Trigonometry problem concerning compound angles

  1. #1
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    Trigonometry problem concerning compound angles

    The diagram (attached below) shows a right-angled triangle ABC in which |AB| = hm
    |<CAD| = 45 deg and |<BAD| = theta

    By using the formula for tan(theta + 45 deg), or otherwise, find the two possible values for h.

    Thanks!

    Diagram:
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  2. #2
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    Re: Trigonometry problem concerning compound angles

    Hello, mathshelp94!


    Code:
          A
          *
          |**
          |@* *
          |  *45*
        h |   *   *
          |    *    *
          |     *     *
          |      *      *
        B *-------*-------* C
          : - 1 - D - 5 - :
    By using the formula for tan(theta + 45 deg), or otherwise, find the two possible values for h.

    \text{We see that: }\:\begin{Bmatrix}\tan\theta \:=\:\dfrac{1}{h} \\ \tan(\theta + 45^o) \:=\:\dfrac{6}{h} \end{Bmatrix}

    \tan(\theta + 45^o) \;=\;\dfrac{\tan\theta + \tan45^o}{1 - \tan\theta\tan45^o} \:=\:\dfrac{6}{h}

    . . . . . . . . . . . . . \dfrac{\frac{1}{h} + 1}{1 -  \frac{1}{h}} \;=\;\dfrac{6}{h}

    Multiply by \frac{h}{h}\!:\quad\;\;\dfrac{1 + h}{h - 1} \;=\;\dfrac{6}{h} \quad\Rightarrow\quad h+h^2 \:=\:6h - 6

    . . h^2 - 5h + 6 \:=\:0 \quad\Rightarrow\quad (h-2)(h-3) \:=\:0 \quad\Rightarrow\quad \boxed{h \:=\:2\text{ or }3}

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