Question

Let
x = 2 sin θ, − π/2 < θ < π/2.
Simplify the expression.
x/square root of 4-x^2

Answer 1

ok i get it
it means replace \(x\) by \(2\sin(\theta)\) right?

Answer 2

\[\frac{x}{\sqrt{4-x^2}}\] first make the direct replacement and get
\[\frac{2\sin(\theta)}{\sqrt{4-(2\sin(\theta)^2}}\]

Answer 3

then square and get
\[\frac{2\sin(\theta)}{\sqrt{4-4\sin^2(\theta)}}\]

Answer 4

factor out the 4 and get
\[\frac{2\sin(\theta)}{\sqrt{4(1-\sin^2(\theta))}}\]

Answer 5

then since \(1-\sin^2(\theta)=\cos^2(\theta)\) you have
\[\frac{2\sin(\theta)}{\sqrt{4\cos^2(\theta)}}=\frac{2\sin(\theta)}{2\cos(\theta)}=\tan(\theta)\]

Answer 6

Thank you :)