Question

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

The $y$-coordinate of $P$ is$\frac{2}{3}$, and the $x$-coordinate is negative.

Short answer

The point$P\left(x,y\right)=\left(-\frac{\sqrt{5}}{3},\frac{2}{3}\right)$.

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 $\frac{2}{3}$ for $y$ into the equation $x^2+y^2=1$ and simplify for $x$.

role="math" localid="1648231010852" $$\begin{gathered} x^2+y^2=1 \\ {\left(x\right)}^2+{\left(\frac{2}{3}\right)}^2=1 \\ {x}^2+\frac{4}{9}-\frac{4}{9}=1-\frac{4}{9} \\ {x}^2=\frac{5}{9} \\ x=\pm \frac{\sqrt{5}}{3} \end{gathered}$$

Step 3. State the conclusion.

Since $x$-coordinate is given to be negative, therefore, out of two calculated values of $x$, the only value that satisfies the condition is $x=-\frac{\sqrt{5}}{3}$.