find theta

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find theta

find theta without the use of the law of cosines

see image attached

Quote:

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!

Quote:

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

Code:

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

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

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

$\begin{array}{ccc}\text{In right triangle }ADB\!: & x^2+y^2 \:=\:5^2 & [1] \\ \\<br /> <br /> \text{In right triangle }ADC\!: & x^2 + (y+3)^2 \:=\:7^2 & [2] \end{array}$

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

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

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