Material Type:
Ohio Open Ed Collaborative
  • Mathematics
  • Tmm0022
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    Creative Commons Attribution Non-Commercial
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    The Six Circular Functions and Fundamental Identities


    TMM 002 PRECALCULUS (revised March 21, 2017)

    2.  Geometry

        2b.  Analyze right triangles.  Routine analysis of side lengths and angle measurements using trigonometric ratios/functions, as well as the Pythagorean Theorem.*

          Sample Tasks:

    • The student can solve right triangles numerically using trigonometric ratios and relationships.
    • The student can compare similar triangles numerically.
    • The student can describe relationships within or between right/similar triangles algebraically using trigonometric ratios and relationships.

    Tangent, Cotangent, Secant and Cosecant Functions and Fundamental Identities

    Having defined the cosine and sine functions, we now define the rest of the circular/trigonometric functions – Tangent, Cotangent, Secant and Cosecant. Next we derive fundamental identities between all six trigonometric function. We note that the values of these functions of an angle, when they exist, are all the same, up to a sign, as that of their corresponding values of its reference angle.

    Review: Cosine and Sine functions and their definitions in terms of the coordinates of points on the unit circle.

    Learning Objectives:

    • Define and understand the tangent, cotangent, secant and cosecant functions in terms of x and y coordinates of a point on the unit circle. Then extend the notion to points on circle of radius r with center at the origin
    • Understand the relationships these functions have with that of Cosine and Sine functions.
    • Reciprocal and Quotient Identities
    • Values of circular functions for any angle in terms of values of the functions for the corresponding reference angle.
    • Pythagorean Identities
    • Pythagorean Conjugates
    • Co-Function Identities
    • Verification of Identities
    • Domain and Range of the circular functions