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The Animated Unit Circle & Trig Functions




Three Triangles of Segments & A Unit Circle

    Each of the six trig functions may be thought of as based on

  • the length of a segment of one of three right triangles,
  • the central angle, x, involved in each of the triangles,
  • the circle with a radius of 1 involved with each triangle.


The Sine and Cosine

    The triangle touching the unit circle at point (x,y) determines the sine and cosine functions.

    The triangle has:
  • a vertical leg, THE SINE FUNCTION, with endpoints at (x,y) - a point on the circle - and (x,0) - a point on the x-axis,
  • a horizontal leg, THE COSINE FUNCTION, with endpoints at (x,0) and the origin, (0,0), and
  • a hypotenuse, length (or radius) of 1, with endpoints at the origin and the point (x,y) on the circle.

    As the angle changes, the sine and cosine change and the radius remains equal to 1.

    Use your imagination and assume the animated figure has a radius of 1.

    Watch the length of the vertical leg of the triangle change as the central angle changes.

    The vertical segment, the vertical leg, is the sine.

    The angle x, because it's in radian measure, cuts the circle at a certain point (x,y) and the vertical length from the x-axis to the point (x,y) is y, the sine of the angle. Said another way, the vertical segment, the y is equal to the sine of the angle, y = sin(x).

    Next, watch the length of the horizontal leg of the triangle change as the central angle changes.

    The horizontal segment, the horizontal leg, is the cosine.

    The sine and the cosine range in value from a low of negative one, -1, to a high of positive 1, +1. No matter how large or small the angle, the functions increase to 1 and decrease to -1.

    The sine and cosine are continuous functions. They have no breaks or holes (undefined values) or asymptotes.

    For a detailed analysis of the sine or cosine, go to The Animated Sine and Cosine.   For detailed information on the sine and cosine functions, go to Function and Relation Library: Trigonometric Functions - sine or cosine.



The Tangent and Secant

    The triangle tangent to the unit circle at the point (1,0), on the x-axis determines the tangent and secant functions.

    The triangle has:
  • a vertical leg, THE TANGENT, the segment with endpoints at (1,0) or (-1,0) and the point of intersection with the secant, if it exists,
  • a horizontal leg, equal to 1, the radius of the circle, and
  • a hypotenuse, THE SECANT, the segment with endpoints at (0,0) and the point of intersection with the tangent, if it exists.

    Below is the animation for the tangent (vertical leg of the triangle) and the secant (hypotnuse of the triangle).

    These functions are more complex than the sine and cosine. Look carefully at the animation and see that both the tangent/leg and the secant/hypotenuse at times are so large that the segment goes off the screen. Each of these functions at times gets infinitely large and can not be displayed.

    When the angle is 0, the tangent is zero and the secant is 1.

    When the angle is pi or 180°, the tangent is zero and the secant is -1

    When the angle is 90° or /2, the tangent is positive infinity, the secant is positive infinity.

    When the angle is is 270° or 3/2, the tangent is at negative infinity, the secant is at negative infinity.

    The range of the tangent is from negative infinity to positive infinity.

    The range of the secant is from 1 to positive infinity and from -1 to negative infinity.

    For a detailed analysis of the tangent or secant, go to The Animated Tangent and Secant.   For detailed information on the tangent and secant functions, go to Function and Relation Library: Trigonometric Functions - tangent or secant.



The Cotangent and Cosecant

    The triangle tangent to the unit circle at the point (0,1), or (0,-1) on the y-axis determines the cotangent and cosecant functions.

    The triangle has:
  • a vertical leg, equal to 1, the radius of the circle, and
  • a horizontal leg, THE COTANGENT, the segment with endpoints at (0,1) or (0,-1) and the point of intersection with the cosecant, if it exists, and
  • a hypotenuse, THE COSECANT, the segment with endpoints at (0,0) and the point of intersection with the cotangent, if it exists.

    Below is the animation for the cotangent (horizontal leg) at (0,1) and the cosecant (hypotnuse).

    As with the tangent and secant, the cotangent and cosecant are more complex than the sine and cosine. Look carefully at the animation and see that both the cotangent/leg and the cosecant/hypotenuse at times are so large that the segment goes off the screen. Each of these functions at times gets infinitely large and can not be displayed.

    When the angle is 0, the cotangent is infinitely large and the cosecant is infinitely large.

    When the angle is pi or 180°, the cotangent is infinitley large and the cosecant is infinitely large.

    When the angle is 90° or /2, the cotangent is 0, the cosecant is 1.

    When the angle is is 270° or 3/2, the cotangent is 0, the cosecant is -1.

    The range of the cotangent is from negative infinity to positive infinity.

    The range of the cosecant is from 1 to positive infinity and from -1 to negative infinity.

    For a detailed analysis of the sine or cosine, go to The Animated Cosecant and Cotangent.   For detailed information on the sine and cosine functions, go to Function and Relation Library: Trigonometric Functions - cotangent or cosecant.




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