Question

Look at the equation [x/square root of (1 + x^2)] = (4/5)
a. Make the subsitution x = tan(θ) in the above equation. Simplify using trig identities and solve for θ.
b. Use this value of θ to find the x value that satisfies the original equation.

Answer 1

\[\frac{x}{\sqrt{1+x^2}}=\frac{4}{5}\]
so we have \[x=\tan(\theta)=> x^2=\tan^2(\theta) => x^2+1=\tan^2(\theta)+1\]
\[\frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}}=\frac{4}{5}\]
\[\frac{\tan(\theta)}{\sqrt{\sec^2(\theta)}}=\frac{4}{5}\]
\[\frac{\tan(\theta)}{\sec(\theta)}=\frac{4}{5}\]
\[\tan(\theta) \div \sec(\theta)=\frac{4}{5}\]
\[\frac{\sin(\theta)}{\cos(\theta)} \div \frac{1}{\cos(\theta)}=\frac{4}{5}\]
\[\frac{\sin(\theta)}{\cos(\theta)}*\frac{\cos(\theta)}{1}=\frac{4}{5}\]
\[\sin(\theta)=\frac{4}{5}\]
this should help