Answer:
[tex]\bold{x =cos\theta}\\\bold{y=sin\theta}[/tex]
Step-by-step explanation:
The given system of linear equations is:
[tex](cos\theta)x+(sin\theta)y=1\\(-sin\theta)x+(cos\theta)y=0[/tex]
We have to solve the equations for the values of [tex]x, y[/tex].
Let us use elimination method in which we eliminate one of the variables from the two variables.
For this, let us multiply the first equation by [tex]sin\theta[/tex] and second equation by [tex]cos\theta[/tex]
Now, the equations become:
[tex](cos\theta.sin\theta)x+(sin\theta.sin\theta)y=sin\theta\\\Rightarrow (cos\theta.sin\theta)x+(sin^2\theta)y=sin\theta ....... (1)\\\\(-sin\theta.cos\theta)x+(cos\theta.cos\theta)y=0\\\Rightarrow (-sin\theta.cos\theta)x+(cos^2\theta)y=0 ..... (2)[/tex]
Now, let us add (1) and (2):
[tex](sin^2\theta)y+(cos^2\theta)y=sin\theta\\\Rightarrow (sin^2\theta+cos^2\theta)y=sin\theta\\\Rightarrow (1)y=sin\theta\\\Rightarrow y = sin\theta[/tex]
Using the equation:
[tex](-sin\theta)x+(cos\theta)y=0[/tex]
Putting value of [tex]y[/tex]:
[tex]\Rightarrow (-sin\theta)x+(cos\theta)sin\theta=0\\\Rightarrow (sin\theta)x=(cos\theta)sin\theta\\\Rightarrow x = cos\theta[/tex]
So, the answer to the system of linear equations is:
[tex]\bold{x =cos\theta}\\\bold{y=sin\theta}[/tex]
