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Find a simplified expression for sin ( tan 1 ( 4 x ) ) for 1 4 x 1 4 .

4 x 16 x 2 + 1

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Key concepts

  • An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
  • Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
  • For any trigonometric function f ( x ) , if x = f 1 ( y ) , then f ( x ) = y . However, f ( x ) = y only implies x = f 1 ( y ) if x is in the restricted domain of f . See [link] .
  • Special angles are the outputs of inverse trigonometric functions for special input values; for example, π 4 = tan 1 ( 1 ) and π 6 = sin 1 ( 1 2 ) . See [link] .
  • A calculator will return an angle within the restricted domain of the original trigonometric function. See [link] .
  • Inverse functions allow us to find an angle when given two sides of a right triangle. See [link] .
  • In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, sin ( cos 1 ( x ) ) = 1 x 2 . See [link] .
  • If the inside function is a trigonometric function, then the only possible combinations are sin 1 ( cos x ) = π 2 x if 0 x π and cos 1 ( sin x ) = π 2 x if π 2 x π 2 . See [link] and [link] .
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See [link] .
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See [link] .

Section exercises


Why do the functions f ( x ) = sin 1 x and g ( x ) = cos 1 x have different ranges?

The function y = sin x is one-to-one on [ π 2 , π 2 ] ; thus, this interval is the range of the inverse function of y = sin x , f ( x ) = sin 1 x . The function y = cos x is one-to-one on [ 0 , π ] ; thus, this interval is the range of the inverse function of y = cos x , f ( x ) = cos 1 x .

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Since the functions y = cos x and y = cos 1 x are inverse functions, why is cos 1 ( cos ( π 6 ) ) not equal to π 6 ?

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Explain the meaning of π 6 = arcsin ( 0.5 ) .

π 6 is the radian measure of an angle between π 2 and π 2 whose sine is 0.5.

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Most calculators do not have a key to evaluate sec 1 ( 2 ) . Explain how this can be done using the cosine function or the inverse cosine function.

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Why must the domain of the sine function, sin x , be restricted to [ π 2 , π 2 ] for the inverse sine function to exist?

In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [ π 2 , π 2 ] so that it is one-to-one and possesses an inverse.

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Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
no. should be about 150 minutes.
It should be 158.5 minutes.
ok, thanks
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
yeah, it does. why do we attempt to gain all of them one side or the other?
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro; thanks for putting it out there like that, 😁
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
dont forget the cube in each variable ;)
of she solves that, well ... then she has a lot of computational force under her command ....
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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