Question

Find the missing coordinate of $P$, using the fact that $P$ lies on the unit circle in the given quadrant.

Coordinates Quadrant

$P\left(\frac{2}{5},\begin{array}{c}\end{array}\right)$ I

Short answer

The missing coordinate of point$P$ is$y=\frac{\sqrt{21}}{5}$.

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="1648209270254" $$\begin{gathered} x^2+y^2=1 \\ {\left(\frac{2}{5}\right)}^2+{\left(y\right)}^2=1 \\ \frac{4}{25}+y^2-\frac{4}{25}=1-\frac{4}{25} \\ {y}^2=\frac{21}{25} \\ y=\pm \frac{\sqrt{21}}{5} \end{gathered}$$

Step 3. State the conclusion.

Since the point is in quadrant I, therefore, $y$-coordinate of the point must be positive and hence, out of two calculated values of $y$, the only value that satisfies the condition is $y=\frac{\sqrt{21}}{5}$.