unit circle

(noun)

A circle centered at the origin with radius 1.

Related Terms

Examples of unit circle in the following topics:

  • 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.

  • 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.
    • Cotangent can also be calculated with values in the unit circle.
    • We now recognize six trigonometric functions that can be calculated using values in the unit 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.

  • Angle Addition and Subtraction Formulae

    • We can use the special angles, which we can review in the unit circle shown below.
    • To see how these formulae are derived, we can place points on a diagram of a unit circle.
    • Substitute the values of the trigonometric functions from the unit circle:
    • Substitute the values of the trigonometric functions from the unit circle:
    • The unit circle with the values for sine and cosine displayed for special angles.

  • Pythagorean Identities

    • Using the definitions of sine and cosine, we will learn how they relate to each other and the unit circle.
    • For any point on the unit circle,
    • For a triangle drawn inside a unit circle, the length of the hypotenuse of the triangle is equal to the radius of the circle, which is $1$.
    • Because $x = \cos t$ and $y= \sin t$ on the unit circle, we can substitute for $x$ and $y$ to get the Pythagorean identity:
    • For a triangle drawn inside a unit circle, the length of the hypotenuse is equal to the radius of the circle.

  • Radians

    • To find another unit, think of the process of drawing a circle.
    • The radian is the standard unit used to measure angles in mathematics.
    • Note that when an angle is described without a specific unit, it refers to radian measure.
    • A unit circle is a circle with a radius of 1, and it is used to show certain common angles.
    • Explain the definition of radians in terms of arc length of a unit circle and use this to convert between degrees and radians

  • Unit Vectors and Multiplication by a Scalar

    • A useful concept in the study of vectors and geometry is the concept of a unit vector.
    • A unit vector is a vector with a length or magnitude of one.
    • The unit vectors are different for different coordinates.
    • The unit vectors in Cartesian coordinates describe a circle known as the "unit circle" which has radius one.
    • If you were to draw a line around connecting all the heads of all the vectors together, you would get a circle of radius one.

  • Angular Position, Theta

    • The angle of rotation is a measurement of the amount (the angle) that a figure is rotated about a fixed point— often the center of a circle.
    • We know that for one complete revolution, the arc length is the circumference of a circle of radius r.
    • The circumference of a circle is 2πr.
    • This result is the basis for defining the units used to measure rotation angles to be radians (rad), defined so that:
    • The radius of a circle is rotated through an angle Δ.

  • Simple Harmonic Motion and Uniform Circular Motion

    • The distance of the body from the center of the circle remains constant at all times.
    • For a path around a circle of radius r, when an angle θ (measured in radians) is swept out, the distance traveled on the edge of the circle is s = rθ.
    • The point P travels around the circle at constant angular velocity ω.
    • The velocity of the point P around the circle equals |vmax|.
    • The angular velocity ω is in radians per unit time; in this case 2π radians is the time for one revolution T.