6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions Flashcards Preview

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

Flashcards in 6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions Deck (12)
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1
Q

The Inverse Secant, Cosecant, and Cotangent Functions

A

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

2
Q

note

A
  • 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.
3
Q

Which trigonometric function is represented by the following graph?

A

cosecant

4
Q

Which inverse trigonometric function is represented by the following graph?

A

arccotangent

5
Q

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

A

The function shown is one-to-one.

6
Q

Which inverse trigonometric function is represented by the following graph?

A

arccosecant

7
Q

Which trigonometric function is represented by the following graph?

A

cotangent

8
Q

Which inverse trigonometric function is represented by the following graph?

A

arcsecant

9
Q

Which trigonometric function is represented by the following graph?

A

secant

10
Q

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

A

The function shown is one-to-one.

11
Q

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

A

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

12
Q

Which inverse trigonometric function is represented by the following graph?

A

arcsine

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