6.5.1 The Inverse Sine, Cosine, and Tangent Functions Flashcards Preview

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

Flashcards in 6.5.1 The Inverse Sine, Cosine, and Tangent Functions Deck (12):
1

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.”

2

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.

3

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)

4

Which of the following is not true?

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

5

Which of the following is not defined?

arcsin (1.2)

6

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

The function arccos x is increasing.

7

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

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

8

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)

9

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.

10

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 )

11

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

12

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.

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