Question

Given that x=3sin​​θ-2 and y=3cos​​​θ+4.Show that (x+2)²+ (y-4)²=9​​​​​​

Answer 1

First, plug-in x and y as 3sin​​θ-2 and 3cos​​​θ+4 into the equation, respectively:

`((3\sin (\theta)-2)+2)^2 + ((3\cos (\theta)+4)-4)^2 = 9`

Then, +2 and -2 cancel out and +4 and -4 cancel out as well, leaving you with:

`(3\sin(\theta))^2+(3\cos(\theta))^2=9`

We can factor out 3^2 = 9 from both equations:

`9(\sin(\theta)^2+cos(\theta)^2) = 9`

We know from a trigonometric identity that
`\sin(\theta)^2+cos(\theta)^2 = 1`, meaning we can reduce the equation to:

`9(1) = 9`

`9=9`

And therefore, we have shown that (x+2)^2 + (y-4)^2 = 9, if x=3sin​​θ-2 and y=3cos​​​θ+4.

Hope this helped you.