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Back to FlashcardsMathematics / AP Calculus AB: 6.5.1 The Inverse Sine, Cosine, and Tangent Functions

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

Mathematics12 CardsCreated 12 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.