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AP Calculus AB: 6.5.1 The Inverse Sine, Cosine, and Tangent Functions

Mathematics12 CardsCreated 3 months ago

This section explains how inverse trigonometric functions—arcsin, arccos, and arctan—are defined by restricting the domains of the original sine, cosine, and tangent functions to make them one-to-one. It covers their notations, domain restrictions, and key properties, including common misconceptions and comparisons of their values.

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The Inverse Sine, Cosine, and Tangent Functions

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

  • The inverse trig functions can be indicated by a raised –1 or by the prefix “arc.”

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Key Terms

Term
Definition

The Inverse Sine, Cosine, and Tangent Functions

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

    • <...

note

  • The sine, cosine, and tangent functions do not pass the
    horizontal line test. Therefore, they do not have inverses.

  • Despi...

Put the following expressions in order of value from the largest to the smallest. arctan(1/2), arctan(−2), 2tan(π/4), arctan(0)

2tan(π/4), arctan(1/2), arctan(0), arctan(−2)

Which of the following is not true?

The sine function is invertible, and arcsin x is the inverse function.

Which of the following is not defined?

arcsin (1.2)

Which of the following statements about arccos x is not true?

The function arccos x is increasing.

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AP Calculus AB: 6.5.1 The Inverse Sine, Cosine, and Tangent Functions
TermDefinition

The Inverse Sine, Cosine, and Tangent Functions

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

  • The inverse trig functions can be indicated by a raised –1 or by the prefix “arc.”

note

  • The sine, cosine, and tangent functions do not pass the
    horizontal line test. Therefore, they do not have inverses.

  • Despite that fact, it would be useful to find a way to define inverse trigonometric functions.

  • Notice that from –π/2 to π/2 the sine function is increasing. On this restricted domain sine is one-to-one.

  • You can find other domains where sine is increasing and therefore one-to-one. You can even find domains where it is decreasing. The sine function will be one-to-one there too. You can define lots of inverses for the sine function. However, mathematicians established a convention to use [–π/2, π/2] for the standard inverse sine function.

  • Here are the graphs of the inverses of sine, cosine, and
    tangent. They are labeled arcsine, arccosine, and
    arctangent, respectively, instead of using the –1 notation.

  • Notice that arccosine is defined by a different interval than the others. The cosine function is restricted to the interval [0, π] in order to define its inverse.

  • Tangent is restricted to [–π/2, π/2], just like sine. The vertical asymptotes for tangent are translated into horizontal asymptotes for arctangent.

Put the following expressions in order of value from the largest to the smallest. arctan(1/2), arctan(−2), 2tan(π/4), arctan(0)

2tan(π/4), arctan(1/2), arctan(0), arctan(−2)

Which of the following is not true?

The sine function is invertible, and arcsin x is the inverse function.

Which of the following is not defined?

arcsin (1.2)

Which of the following statements about arccos x is not true?

The function arccos x is increasing.

Which of the following statements about arctan x is not true?

The domain of definition for arctan x is −1 ≤ x ≤ 1.

Put the following expressions in order of value from the smallest to the largest:
arccos (1/2), arccos (−1/3), −e^ −2, arccos (0)

−e^ −2, arccos (1/2), arccos (0), arccos (−1/3)

Which of the following is the graph of y = arccos x ?

This is the graph of y = arccos x, which is the reflection of the graph of y = cos x on the restricted domain over the line given by
y = x. This graph matches the domain and range-of-values conditions.

Put the expressions in order of value from the smallest to the largest.
arcsin (1/2), arcsin (1/3), ln (e^2 ), arcsin (0)

arcsin (0), arcsin (1/3), arcsin (1/2), ln (e^2 )

Which of the following is the graph of y = arctan x ?

This is the graph of y = arctan x, which is the reflection of the graph of y = tan x on the restricted domain over the line given by y = x. This graph matches the domain and range-of-values conditions

Which of the following is the graph of y = arcsin x ?

This is the graph of y = arcsin x, which is the reflection of the graph of y = sin x on the restricted domain over the line given by
y = x. This graph matches the domain and range-of-values conditions for y = arcsin x.