2.2 The Unit Circle (continued)

Summary

Let $(x,y)$ be a point on the unit circle. In each of the following situations, determine the requested value exactly.

• Suppose that $x = -0.3$ and $y$ is negative. Find the value of $y$. - Suppose that $(x,y)$ lies in Quadrant II and $x = -2y$. Find the values of $x$ and $y$. - Suppose that $(x,y)$ lies a distance of $\frac{29\pi}{6}$ units clockwise around the circle from $(1,0)$. Find the values of $x$ and $y$. - At what exact point(s) does the line $y = \frac{1}{2}x + \frac{1}{2}$ intersect the unit circle?

The unit circle is centered at $(0,0)$ and radius $r = 1$, from which the Pythagorean Theorem tells us that any point $(x,y)$ on the unit circle satisfies the equation $x^2 + y^2 = 1$.

• Explain why any point $(x,y)$ on a circle of radius $r$ centered at $(h,k)$ satisfies the equation $(x-h)^2 + (y-k)^2 = r^2$. - Determine the equation of a circle centered at $(-3,5)$ with radius $r = 2$. - Suppose that the unit circle is magnified by a factor of $5$ and then shifted $4$ units right and $7$ units down. What is the equation of the resulting circle? - What is the length of the arc intercepted by a central angle of $\frac{2\pi}{3}$ radians in the circle $(x-1)^2 + (y-3)^2 = 16$? - Suppose that the line segment from $(-2,-1)$ to $(4,2)$ is a diameter of a circle. What is the circle's center, radius, and equation?

Consider the circle whose center is $(0,0)$ and whose radius is $r = 5$. Let a point $(x,y)$ traverse the circle counterclockwise from $(5,0)$, and say the distance along the circle from $(5,0)$ is represented by $d$.

• Consider the point $(a,b)$ that is generated by the central angle $\theta$ with vertices $(5,0)$, $(0,0)$, and $(a,b)$. If $\theta = \frac{\pi}{6}$, what are the exact values of $a$ and $b$? - Answer the same question as in (a) except with $\theta = \frac{\pi}{4}$ and $\theta = \frac{\pi}{3}$. - How far has the point $(x,y)$ traveled after it has traversed the circle one full revolution? - Let $h = f(d)$ be the circular function that tracks the height of the point $(x,y)$ as a function of distance, $d$, traversed counterclockwise from $(5,0)$. Sketch an accurate graph of $f$ through two full periods, labeling several special points on the graph as well as the horizontal and vertical scale of the axes.