find theta without the use of the law of cosines
see image attached
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: .$\displaystyle \theta \;=\;\frac{\pi}{3}\;\;(60^o)$