Question

The point $P$ is on the unit circle. Find$P\left(x,y\right)$from the given information.

The $x$-coordinate of $P$ is positive, and the $y$-coordinate of $P$ is$-\frac{\sqrt{5}}{5}$.

Short answer

The point$P\left(x,y\right)=\left(\frac{2\sqrt{5}}{5},-\frac{\sqrt{5}}{5}\right)$.

Step-by-step solution

Step 1. State the definition of the unit circle.

The unit circle is the circle of radius $1$ centered at the origin in the $xy$- plane. It is given by the equation $x^2+y^2=1$.

Step 2. Substitute the given points and simplify for x.

Since the point satisfies the equation of the unit circle, substitute$-\frac{\sqrt{5}}{5}$ for $y$ into the equation $x^2+y^2=1$and simplify for $x$.

localid="1648232225987" $$\begin{gathered} x^2+y^2=1 \\ {\left(x\right)}^2+{\left(-\frac{\sqrt{5}}{5}\right)}^2=1 \\ {x}^2+\frac{5}{25}-\frac{5}{25}=1-\frac{5}{25} \\ {x}^2=\frac{20}{25} \\ x=\pm \frac{2\sqrt{5}}{5} \end{gathered}$$

Step 3. State the conclusion.

Since $x$-coordinate is given to be positive, therefore, out of two calculated values of $x$, the only value that satisfies the condition is $x=\frac{2\sqrt{5}}{5}$.