circle

(noun)

The conic section formed by the plane being parallel to the base of the cone.

Related Terms

  • diameter
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • focus
  • eccentricity
  • Parabola
  • ellipse
  • hyperbola
  • asymptote
  • area
  • radius
  • vertex

(noun)

A two-dimensional geometric figure, consisting of the set of all those points in a plane that are equally distant from another point.

Related Terms

  • diameter
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • focus
  • eccentricity
  • Parabola
  • ellipse
  • hyperbola
  • asymptote
  • area
  • radius
  • vertex

(noun)

A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.

Related Terms

  • diameter
  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • focus
  • eccentricity
  • Parabola
  • ellipse
  • hyperbola
  • asymptote
  • area
  • radius
  • vertex

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.
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