Trigonometry problem concerning compound angles

**mathshelp94** · March 5th 2012, 01:02 PM

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: Image - TinyPic - Free Image Hosting, Photo Sharing & Video Hosting

**Soroban** · March 5th 2012, 08:24 PM

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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