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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

Verbal

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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
Jude
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what is chemistry
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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