circle

Examples of circle in the following topics:

• Circles as Conic Sections

  • You've known all your life what a circle looks like.
  • But what is the exact mathematical definition of a circle?
  • Hence, the definition for a circle as given above.
  • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
  • Tangent: a straight line that touches the circle at a single point.

• Introduction to Circles

  • The equation for a circle is an extension of the distance formula.
  • The definition of a circle is as simple as the shape.
  • Since we know a circle is the set of points a fixed distance from a center point, let's look at how we can construct a circle in a Cartesian coordinate plane with variables $x$ and $y$.
  • Now that we have an algebraic foundation for the circle, let's connect it to what we already know about some different parts of the circle.
  • The circumference is the length of the path around the circle.

• Defining Trigonometric Functions on the Unit Circle

  • In this section, we will redefine them in terms of the unit circle.
  • Recall that a unit circle is a circle centered at the origin with radius 1.
  • The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram.
  • We can find the coordinates of any point on the unit circle.
  • The unit circle demonstrates the periodicity of trigonometric functions.

• Radians

  • An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation.
  • The length of the arc around an entire circle is called the circumference of that circle.
  • One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.
  • A unit circle is a circle with a radius of 1, and it is used to show certain common angles.
  • An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian.

• Introduction to Ellipses

  • Then we can write the equation of the circle in this way:
  • In this equation, $r$ is the radius of the circle.
  • A circle has only one radius—the distance from the center to any point is the same.
  • To change our circle into an ellipse, we will have to stretch or squeeze the circle so that the distances are no longer the same.
  • First, let's start with a specific circle that's easy to work with, the circle centered at the origin with radius $1$.

• Applications of Circles and Ellipses

  • Therefore the equation of this circle is:
  • The center of the circle can be found by comparing the equation in this exercise to the equation of a circle:
  • The radius of the circle is $r$.
  •  The leftmost point on the circle is $(-3,-8)$.
  • The radius of the circle is $r$.

• Secant and the Trigonometric Cofunctions

  • Trigonometric functions have reciprocals that can be calculated using the unit circle.
  • It is easy to calculate secant with values in the unit circle.
  • As with secant, cosecant can be calculated with values in the unit circle.
  • Recall that for any point on the circle, the $y$-value gives $\sin t$.
  • Cotangent can also be calculated with values in the unit circle.

• Types of Conic Sections

  • A circle is formed when the plane is parallel to the base of the cone.
  • All circles have certain features:
  • All circles have an eccentricity $e=0$.
  • On a coordinate plane, the general form of the equation of the circle is
  • The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.

• Sine and Cosine as Functions

  • The functions sine and cosine can be graphed using values from the unit circle, and certain characteristics can be observed in both graphs.
  • Recall that the sine and cosine functions relate real number values to the $x$- and $y$-coordinates of a point on the unit circle.
  • Notice how the sine values are positive between $0$ and $\pi$, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between $\pi$ and $2\pi$, which correspond to the values of the sine function in quadrants III and IV on the unit circle.
  • The points on the curve $y = \sin x$ correspond to the values of the sine function on the unit circle.
  • The points on the curve $y = \cos x$ correspond to the values of the cosine function on the unit circle.

• Special Angles

  • The unit circle and a set of rules can be used to recall the values of trigonometric functions of special angles.
  • The angles identified on the unit circle above have relatively simple expressions.
  • Note that while only sine and cosine are defined directly by the unit circle, tangent can be defined as a quotient involving these two:
  • Applying rules and shortcuts associated with the unit circle allows you to solve trigonometric functions quickly.
  • Special angles and their coordinates are identified on the unit circle.