6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions

Question 1

The Inverse Secant, Cosecant, and Cotangent Functions

Answer 1

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

Question 2

note

Answer 2

• 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.

Question 3

Which trigonometric function is represented by the following graph?

Answer 3

cosecant

Question 4

Which inverse trigonometric function is represented by the following graph?

Answer 4

arccotangent

Question 5

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

Answer 5

The function shown is one-to-one.

Question 6

Which inverse trigonometric function is represented by the following graph?

Answer 6

arccosecant

Question 7

Which trigonometric function is represented by the following graph?

Answer 7

cotangent

Question 8

Which inverse trigonometric function is represented by the following graph?

Answer 8

arcsecant

Question 9

Which trigonometric function is represented by the following graph?

Answer 9

secant

Question 10

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

Answer 10

The function shown is one-to-one.

Question 11

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

Answer 11

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

Question 12

Which inverse trigonometric function is represented by the following graph?

Answer 12

arcsine