# Math Help - ladder with angle

A 12 foot solar panel is to be installed on a roof with a 15 degree pitch. Find the length of the vertical brace d if the panel must be installed to make a 40 degree angle with the horizontal.

I guess what i really don't understand if where the 40 degree angle is....

2. Hello, kelsey3!

A 12-foot solar panel is to be installed on a roof with a 15° pitch.
Find the length of the vertical brace $d$ if the panel
must make a 40° angle with the horizontal.
Code:
                              * B
*   |d
12   *       * C
*     *     |
*   *           |
* * 15°             |
A * - - - - - - - - - - - * D

The solar panel is: . $AB = 12$
$\angle CAD = 15^o,\;\angle BAD = 40^o$

In right triangle $BDA\!:\;\begin{array}{cccccccc}\sin40^o &=& \frac{BD}{12} &\Rightarrow& BD &=& 12\sin40^o& {\color{blue}[1]}\\ \\[-4mm]
\cos40^o &=& \frac{AD}{12} & \Rightarrow & AD & =& 12\cos40^o & {\color{blue}[2]}\end{array}$

In right triangle $CDA\!:\;\;\tan15^o \,=\,\frac{CD}{AD}\quad\Rightarrow\quad CD \:=\:AD\tan15^o \;{\color{blue}[3]}$

Substitute [2] into [3]: . $CD \:=\:12\cos40^o\tan15^o\;{\color{blue}[4]}$

We have: . $d \;=\;BC \;=\;BD - CD\;{\color{blue}[5]}$

Substitute [1] and [4] into [5]: . $\boxed{d \;=\;12\sin40^o - 12\cos40^o\tan15^o}$