Question

The point $P$ is on the unit circle. Findrole="math" localid="1648233498232" $P\left(x,y\right)$from the given information.

The $x$-coordinate of $P$ is$-\frac{2}{5}$, and $P$ lies above the $x$-axis.

Short answer

The point$P\left(x,y\right)=\left(-\frac{2}{5},\frac{\sqrt{21}}{5}\right)$.

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 y .

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

role="math" localid="1648233753289" $$\begin{gathered} x^2+y^2=1 \\ {\left(-\frac{2}{5}\right)}^2+{\left(y\right)}^2=1 \\ \frac{4}{25}+y^2-\frac{4}{25}=1-\frac{4}{25} \\ {y}^2=\frac{21}{25} \\ y=\pm \frac{\sqrt{21}}{5} \end{gathered}$$

Step 3. State the conclusion.

Since $P$ lies above the $x$-axis, so, $y$-coordinate must be positive, therefore, out of two calculated values of $y$, the only value that satisfies the condition is $y=\frac{\sqrt{21}}{5}$.