Thread: find theta

find theta without the use of the law of cosines

see image attached

Originally Posted by calc123

find theta without the use of the law of cosines

see image attached

Why can't you use the Cosine Rule?

well my prof told me i didnt need to know that for the test. so im trying to solve it a diff way

Hello, calc123!

Find $\displaystyle \theta$ without the use of the Law of Cosines.

Code:

      |     x     A
    D + - - - - - o
      |         **
      |     5 * *
    y |     *  *
      | θ *   *
      | *    *
    B o     * 7
      |    *
      |   *
    3 |  *
      | *
      |*
    C o

We have: .$\displaystyle AB = 5,\;\;BC = 3,\;\;AC = 7$

Let: .$\displaystyle AD = x,\;\;BD = y,\;\;\theta = \angle DBA$


$\displaystyle \begin{array}{ccc}\text{In right triangle }ADB\!: & x^2+y^2 \:=\:5^2 & [1] \\ \\

\text{In right triangle }ADC\!: & x^2 + (y+3)^2 \:=\:7^2 & [2] \end{array}$

Subtract [2] - [1]: .$\displaystyle (y+3)^2 - y^2 \:=\:49 - 25 \quad\Rightarrow\quad 6y \:=\:15 \quad\Rightarrow\quad y \:=\:\frac{5}{2}$

Hence: .$\displaystyle \cos\theta \:=\:\frac{\frac{5}{2}}{5} \:=\:\frac{1}{2}$


Therefore: .$\displaystyle \theta \;=\;\frac{\pi}{3}\;\;(60^o)$