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AP Calculus AB: 6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions

Mathematics12 CardsCreated 3 months ago

This section discusses how the inverse functions of secant, cosecant, and cotangent are defined by restricting the domains of their original trigonometric counterparts to make them one-to-one. It clarifies the difference between reciprocals and inverses and explains the domain and range conventions for each inverse trig function to ensure they remain valid functions.

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The Inverse Secant, Cosecant, and Cotangent Functions

The standard trigonometric functions do not have inverses. Only by restricting the domain can you make them one-to-one functions.

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Key Terms

Term
Definition

The Inverse Secant, Cosecant, and Cotangent Functions

The standard trigonometric functions do not have inverses. Only by restricting the domain can you make them one-to-one functions.

note

  • The cosecant, secant, and cotangent functions are reciprocals
    of the sine, cosine, and cotangent functions.

  • Don’t confuse...

Which trigonometric function is represented by the following graph?

cosecant

Which inverse trigonometric function is represented by the following graph?

arccotangent

For csc x to have an inverse, it should be restricted in which of the following ways?

The function shown is one-to-one.

Which inverse trigonometric function is represented by the following graph?

arccosecant

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AP Calculus AB: 6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions
TermDefinition

The Inverse Secant, Cosecant, and Cotangent Functions

The standard trigonometric functions do not have inverses. Only by restricting the domain can you make them one-to-one functions.

note

  • The cosecant, secant, and cotangent functions are reciprocals
    of the sine, cosine, and cotangent functions.

  • Don’t confuse reciprocal with inverse.

  • Before defining an inverse for cosecant, you must restrict its
    domain. The convention is to restrict it to the interval
    [–pi/2, pi/2].

  • Since cosecant is not defined at zero, arccosecant never equals
    zero. It ranges from –pi /2 to pi/2, skipping zero.

  • Like cosine, secant must be restricted to the interval
    [0, π] in order for it to be invertible.

  • Reflecting the graph of secant across the line defined by y = x produces the graph of arcsecant.

  • Notice that arcsecant never equals defined there.
    pi/2, since secant is not

  • For cotangent, the convention is to restrict the domain to (–pi/2, pi/2]. Notice that this is a half-open interval. This is because cotangent is equal to zero at both –pi/2 and pi/2. You don’t want arccotangent to have both of those values at zero.

  • Notice also that arccotangent never equals zero. This is
    because cotangent is not defined there.

Which trigonometric function is represented by the following graph?

cosecant

Which inverse trigonometric function is represented by the following graph?

arccotangent

For csc x to have an inverse, it should be restricted in which of the following ways?

The function shown is one-to-one.

Which inverse trigonometric function is represented by the following graph?

arccosecant

Which trigonometric function is represented by the following graph?

cotangent

Which inverse trigonometric function is represented by the following graph?

arcsecant

Which trigonometric function is represented by the following graph?

secant

For secx to have an inverse, it should be restricted in which of the following ways?

The function shown is one-to-one.

Which of these graphs does not show an acceptable restricted domain of cot x that would allow it to have an inverse?

On this interval, cot x does not pass the horizontal line test. Therefore, it is not invertible on the shown interval.

Which inverse trigonometric function is represented by the following graph?

arcsine