# Math Help - solving trigonometric equations with 2 variables

1. ## solving trigonometric equations with 2 variables

I finalized the question into 2 equations:

$-xcos70+ycos50-5.4=0$...a
$xsin70+ysin50-23.9=0$...b

Just wondering, for a highschool grade 6th student or a first year university student, what is the fastest way to solve for $x$ and $y$?

I used matrix but it does not seem to be ideal, please see attachment.

2. ## Re: solving trigonometric equations with 2 variables

Originally Posted by zxe
I finalized the question into 2 equations:

$-xcos70+ycos50-5.4=0$...a
$xsin70+ysin50-23.9=0$...b

Just wondering, for a high school grade 6th student or a first year university student, what is the fastest way to solve for $x$ and $y$?

I used matrix but it does not seem to be ideal, please see attachment.
Multiply equation (a) by sin(70°) and equation (b) by cos(70°), then add the two equations and solve for y.

3. ## Re: solving trigonometric equations with 2 variables

Originally Posted by zxe
I finalized the question into 2 equations:

$-xcos70+ycos50-5.4=0$...a
$xsin70+ysin50-23.9=0$...b

Just wondering, for a highschool grade 6th student or a first year university student, what is the fastest way to solve for $x$ and $y$?

I used matrix but it does not seem to be ideal, please see attachment.
Matrices should work ok...

\displaystyle \begin{align*} \left[\begin{matrix}-\cos{70^{\circ}} & \cos{50^{\circ}} \\ \phantom{-}\sin{70^{\circ}} & \sin{50^{\circ}}\end{matrix}\right]\left[\begin{matrix}x \\ y\end{matrix}\right] &= \left[\begin{matrix}5.4 \\ 23.9\end{matrix}\right] \\ \left[\begin{matrix}x \\ y\end{matrix}\right] &= \left[\begin{matrix}-\cos{70^{\circ}} & \cos{50^{\circ}} \\ \phantom{-}\sin{70^{\circ}} & \sin{50^{\circ}}\end{matrix}\right]^{-1}\left[\begin{matrix}5.4 \\ 23.9\end{matrix}\right] \end{align*}