6.5.1 The Inverse Sine, Cosine, and Tangent Functions

Question 1

The Inverse Sine, Cosine, and Tangent Functions

Answer 1

• 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.”

Question 2

note

Answer 2

• 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.

Question 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)

Answer 3

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

Question 4

Which of the following is not true?

Answer 4

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

Question 5

Which of the following is not defined?

Answer 5

arcsin (1.2)

Question 6

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

Answer 6

The function arccos x is increasing.

Question 7

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

Answer 7

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

Question 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)

Answer 8

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

Question 9

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

Answer 9

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.

Question 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)

Answer 10

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

Question 11

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

Answer 11

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

Question 12

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

Answer 12

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.