Find out more secrets about circles; information They don't want you to know. Like how circles relate to trigonometry and how you can use one to divorce yourself from a calculator.

Though this powerful Geogebra demonstration does much more, we will be using it to demonstrate how a plane that intersects a double-napped cone can create a variety of 2-D shapes called conic sections, of which a circle is one. Author_Irina Boyadzhiev

Objective 1a

You will be able to define and use the equation of a circle

Just when you thought you knew what a circle was, along comes the conic section. If you take a slice of a cone parallel to its base, what shape do you get? Duration_9_44

Objective 1b

Because one definition is never enough, let's define a circle as a locus of points, then use that definition to derive the equation of a circle. Duration_9_59

Objective 1c

Now that we have derived the equation of a circle, let's use that equation to solve a few circle problems. Duration_8_31

Objective 1d

Since the graph of a circle is not a function, graphing one on your calculator can be tricky. So let's learn the trick. Duration_8_29

Objective 2a

You will be able to complete and use the unit circle

The unit circle is a tiny thing really, but as my mom used to tell me, good things come in small packages. The good thing in this case is the ability to calculate the exact value of sine, cosine, and tangent for 102 different values. In this video, we lay the groundwork for constructing our unit circle by review special right triangles and radian angle measurement. Duration_9_56

Objective 2b

Now that we have the basics down, let's learn how to cleverly place each angle measurement on our unit circle based on the hours on a clock. Hope you like fractions. And clocks. Duration_14_50

Objective 2c

We have completed the angle measures on the unit circle, now it's time to focus on the (x,y) coordinates of the points. For that, we'll need the aforementioned special right triangles. It will be those coordinates, by the way, that will show us how all of this relates to trigonometry. Duration_12_55

Objective 2d

In this final video, we use Geogebra to further investigate properties of the unit circle. I expect a thank you card from your future Pre-Calculus teacher. Duration_15_18

Geometry 9(A) determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems

Geometry 9(B) apply the relationships in special right triangles 30º-60º-90º and 45º-45º-90º and the Pythagorean theorem, including Pythagorean triples, to solve problems

Geometry 10(A) identify shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres, and identify three-dimensional objects generated by rotations of two-dimensional shapes

Geometry 12(D) describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle

Geometry 12(E) show that the equation of a circle with center at the origin and radius r is x^2+y^2=r^2 and determine the equation for the graph of a circle with radius r and center (h, k), (x-h)^2+(y-k)^2=r^2.