The inverse of a trig function With many of the previous elementary - - PowerPoint PPT Presentation

Elementary Functions

Part 4, Trigonometry Lecture 4.6a, Inverse Trig Functions

• Dr. Ken W. Smith

Sam Houston State University

2013

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The inverse of a trig function

With many of the previous elementary functions, we are able to create inverse functions. For example: the inverse of a linear function is another linear function; the inverse of a quadratic function is (with restricted domain) the square root function; the inverse of an exponential function is a logarithm. And so on....

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The inverse of a trig function

Here we examine inverse functions for the six basic trig functions. Recall that if we are going to take a function f(x) and create the inverse function f−1(x) then the function f(x) needs to be one-to-one. We cannot have two different inputs a and b where y = f(a) = f(b) for then we don’t know how to compute f−1(y). Visually, this says that the graph of y = f(x) must pass the horizontal line test. This is a significant problem for the trig functions since the trig functions are periodic and so, given any y-value, there are an infinite number of x-values such that y = f(x). Trig functions badly fail the horizontal line test! We fix this problem by restricting the domain of the trig functions in order to create inverse functions.

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The inverse of a trig function

Let us take a moment to review the inverse function concept. In the past we used the superscript −1 to indicate an inverse function, writing f−1(x) to mean the inverse function of f(x). We continue to do this, writing sin−1 x for the inverse sine function and tan−1 x for the inverse function of tangent. Etc. But there is another common notation for inverse functions in

• trigonometry. It is common to write “arc

” to indicate an inverse function, since the output of an inverse function is the angle (arc) which goes with the trig value. For example, the inverse function of sin(x) is written either sin−1(x) or arcsin(x). In these notes the terms sin−1 x and arcsin x are equivalent.

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The inverse of a trig function

Let’s practice the concept of an inverse function (while reviewing some of

• ur favorite angles.) Find (without a calculator) the exact values of the

following:

1 arccos( √ 2 2 ) 2 arccos(− √ 3 2 ) 3 arcsin( √ 3 2 ) 4 arctan(1) 5 arctan(

√ 3)

6 arctan(−

√ 3) Solutions.

1 Since arccos(x) is the inverse function of cos(x) then we seek here an

angle θ whose cosine is

√ 2 2 . Since cos( π 4 ) = √ 2 2 then arccos( √ 2 2 )

should be π

4 . 2 arccos(− √ 3 2 ) = 5π 6 since cos( 5π 6 ) = − √ 3 2 . 3 arcsin( √ 3 2 ) = π 3 since sin( π 3 ) = √ 3 2 . 4 arctan(1) = π 4 since tan( π 4 ) = 1. 5 Since tan( π 3 ) =

√ 3 then arctan( √ 3) = π

3 . 6 Since tan(− π 3 ) = −

√ 3 then arctan(− √ 3) = − π

3 .

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Restricting the domain of trig functions

Since the trig functions are periodic there are an infinite number of x-values such that y = f(x). We can fix this problem by restricting the domain of the trig functions so that the trig function is one-to-one in that specific domain. For example, the sine function has domain (−∞, ∞) and range [−1, 1]. It is periodic with period 2π and during each period, each output occurs twice.

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Restricting the domain of trig functions

We can make the sine function one-to-one if we restrict the domain to a region (of length π) in which each output occurs exactly once. If we restrict the domain of sine to [− π

2 , π 2 ] then suddenly the previous

graph looks like this.

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Restricting the domain of trig functions

This new restricted sine function is one-to-one; it satisfies the horizontal line test! (Here it is drawn again, with the x-axis stretched out to make it easier to see.)

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Restricting the domain of trig functions

Now we are ready to create an inverse function. The restricted sine function has domain [− π

2 , π 2 ] and range [−1, 1].

If we swap the inputs and outputs we have a new function with domain [−1, 1] and range [− π

2 , π 2 ].

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The arcsine and arcosine functions

The inverse function for sin(x) is called “inverse sine” or “arc sine.” (I will try to use the term “arcsine”, written arcsin(x).) The arcsine function has domain [−1, 1] and range [− π

2 , π 2 ].

It has the property that on the interval [−1, 1], if y = arcsin(x) then x = sin(y). As the sine function takes in an angle and outputs a real number between −1 and 1, the arcsine function takes in a value between −1 and 1 and gives out a corresponding angle. For example, arcsin( 1

2) must be the angle π 6 since sin( π 6 ) = 1 2.

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The inverse of a trig function

In the next presentation, we will look in depth at the inverse functions of the other trig functions. (End)

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Elementary Functions

Part 4, Trigonometry Lecture 4.6b, Inverse functions for cosine, tangent and secant

• Dr. Ken W. Smith

Sam Houston State University

2013

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The Arccosine function

In the previous presentation we created an inverse function for sin x by restricting the domain of sin x to the interval [− π

2 , π 2 ].

Unfortunately the restricted domain choice we made for the sine function doesn’t work for cosine since cosine is not one-to-one on the interval [− π

2 , π 2 ].

Also cosine is nonnegative on this interval and we want to choose a domain that represents all of the range [−1, 1].

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The Arccosine function

For cosine we will instead choose the restricted domain [0, π] so that each

• utput occurs exactly once.

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The Arccosine function

If we exchange x (inputs) with y (outputs) and so reflect that graph across the line y = x we get the graph of the arccosine function below.

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The other inverse trig functions

In the creation of inverse trig functions, we must always restrict the domain of the original trig function to an interval of length π. This means we will choose our angles to fall into two quadrants of the unit circle. We choose those quadrants with the following properties:

1 We always include the first quadrant ([0, π 2 ]) in our domain. 2 The other quadrant is adjacent to the first quadrant, so it is either

Quadrant II or Quadrant IV.

3 We need to make sure that all values of output (including negative

values) are included in the range, so this means the “other” quadrant

• f the domain is Quadrant II for cosine and secant and Quadrant IV

for sine and cosecant.

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The Arctangent function

The tangent function, like all trig functions, is periodic. It has period π.

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The Arctangent function

We restrict the domain of tan x to an interval of period π so that tangent hits each output exactly once. We can do that if we restrict the tangent to [− π

2 , π 2 ].

(For the tangent function we will include negative angles in Quadrant IV so that we don’t cross a place (such as π

2 or − π 2 ) where the tangent is

undefined.)

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The Arctangent function

In the figure below, we hide (in light yellow) the other branches of the tangent function and focus on the interval [− π

2 , π 2 ] where the tangent

function is one-to-one.

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The Arctangent function

Now we are ready to create the inverse function.

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Arcsecant and Arccosecant

There are similar definitions for the restricted domains that allow us to find inverse functions for sec(x) and csc(x). We restrict sec(x) to the same domain as its reciprocal, cos(x), and we restrict csc(x) to the same domain as its reciprocal, sin(x). We have to be a little careful here since sec(x) is undefined at x = π/2 since cos(π/2) is zero. So technically the new domain of sec(x) is not [0, π] but [0, π/2) ∪ (π/2, π]. (We won’t spend much time worrying about these details as long as we understand the mechanism of inverse functions.)

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The inverse of a trig function

In the next presentation, we will work through some problems using inverse trig functions. (End)

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Elementary Functions

Part 4, Trigonometry Lecture 4.6c, Applied Problems with Inverse Trig Functions

• Dr. Ken W. Smith

Sam Houston State University

2013

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Applied Problems with Inverse Trig Functions

A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it touches level ground 360 feet from the base of the tower. What angle does the wire make with the ground? Solution. The wire forms the hypotenuse of a right triangle in which the right angle is at the base of the tower. The 360 feet from the base of the tower to the spot where the guy wire touches the ground forms another side of the triangle with the angle θ between those two sides. So cos θ = 360

1000 = 0.36.

Therefore θ = arccos(0.36) ≈ 1.20253 ≈ 68.9◦ .

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Applied Problems with Inverse Trig Functions

My back yard needs watering. I set up a sprinkler to water all the dry grass near the fence (which is 100 feet long.) The sprinkler is 35 feet from the center of the fence. What angle do I set on the sprinkler head so that the sprinkler goes back and forth along the fence, covering all 100 feet?

• Solution. The sprinkler needs to rotate so that it moves 50 feet in both

directions from the center of the fence. The sprinkler, the center of the fence and one end of the fence form a right triangle (with right angle at the center of the fence) with short sides of lengths 35 feet and 50 feet. Let θ = arctan(50 35) ≈ 0.96 radians ≈ 55◦. This is the angle the sprinkler must cover from the center of the fence to

• ne end. So 2θ ≈ 1.92 radians ≈ 110◦ is the total angle the sprinkler

must rotate since we want the sprinkler to cover the fence from one end to the other.

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Applied Problems with Inverse Trig Functions

Let’s do one more problem similar to one found on WebAssign. Find all angles which satisfy the equation sin θ = 0.2. Solution. The angle θ = arcsin(0.2) ≈ 0.2014 ≈ 11.54◦ is certainly a solution to this

• equation. But this angle is in the first quadrant. Another solution is in the

second quadrant, with reference angle equal to arcsin(0.2). The angle in the second quadrant is π − arcsin(0.2) ≈ 2.9402 ≈ 180◦ − 11.54◦ = 168.46◦. But there are an infinite number of solutions to the equation sin θ = 0.2. Since the sine function is periodic with period 2π, take any solution and add 2π to it to get another solution! So if k is an integer, arcsin(0.2) + 2πk is a solution as is (π − arcsin(0.2)) + 2πk. In set notation, our solution set for the equation sin θ = 0.2 is {arcsin(0.2) + 2πk : k ∈ Z} ∪ {π − arcsin(0.2) + 2πk : k ∈ Z}.

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Inverse Trig Functions

In the next presentation, we will look in further at applications of inverse trig functions (End)

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