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Home , Inverse function, Sine

The inverse of a trig

With many of the previous elementary functions, we are able to create Elementary Functions inverse functions. Part 4, For example: Lecture 4.6a, Inverse Trig Functions the inverse of a linear function is another linear function;

the inverse of a quadratic function is (with restricted domain) the square Dr. Ken W. Smith root function;

Sam Houston State University the inverse of an is a . 2013 And so on....

Smith (SHSU) Elementary Functions 2013 1 / 27 Smith (SHSU) Elementary Functions 2013 2 / 27 The inverse of a trig function The inverse of a trig function

Here we examine inverse functions for the six basic trig functions. Let us take a moment to review the concept. Recall that if we are going to take a function f(x) and create the inverse In the past we used the superscript −1 to indicate an inverse function, function f −1(x) then the function f(x) needs to be one-to-one. writing f −1(x) to mean the inverse function of f(x). We continue to do We cannot have two different inputs a and b where y = f(a) = f(b) for this, writing sin−1 x for the inverse function and tan−1 x for the then we don’t know how to compute f −1(y). inverse function of . Etc.

Visually, this says that the graph of y = f(x) must pass the horizontal line But there is another common notation for inverse functions in test. trigonometry. It is common to write “arc ” to indicate an inverse This is a significant problem for the trig functions since the trig functions function, since the output of an inverse function is the (arc) which are periodic and so, given any y-value, there are an infinite number of goes with the trig value. x-values such that y = f(x). For example, the inverse function of sin(x) is written either sin−1(x) or arcsin(x). Trig functions badly fail the ! In these notes the terms sin−1 x and arcsin x are equivalent. We fix this problem by restricting the domain of the trig functions in order to create inverse functions.

Smith (SHSU) Elementary Functions 2013 3 / 27 Smith (SHSU) Elementary Functions 2013 4 / 27 The inverse of a trig function Restricting the domain of trig functions

Let’s practice the concept of an inverse function (while reviewing some of Since the trig functions are periodic there are an infinite number of our favorite .) Find (without a calculator) the exact values of the x-values such that y = f(x). following: √ 2 We can fix this problem by restricting the domain of the trig functions so 1 arccos( ) 2√ that the trig function is one-to-one in that specific domain. 3 2 arccos(− ) √ 2 3 For example, the sine function has domain (−∞, ∞) and range [−1, 1]. It 3 arcsin( ) 2 is periodic with period 2π and during each period, each output occurs 4 arctan(1) √ twice. 5 arctan( √3) 6 arctan(− 3) Solutions. 1 Since arccos(x) is the inverse√ function of cos(√x) then we seek√ here an 2 π 2 2 angle θ whose cosine is 2 . Since cos( 4 ) = 2 then arccos( 2 ) should be π . √4 √ 3 5π 5π 3 2 arccos(− ) = since cos( ) = − . √ 2 6 6 √ 2 3 π π 3 3 arcsin( 2 ) = 3 since sin( 3 ) = 2 . π π 4 arctan(1) = since tan( ) = 1. 4 √ 4 √ π π 5 SinceSmith (SHSU)tan( ) = 3 then arctan(Elementary Functions3) = . 2013 5 / 27 Smith (SHSU) Elementary Functions 2013 6 / 27 3 √ √3 π π 6 Since tan(− ) = − 3 then arctan(− 3) = − . Restricting the3 domain of trig functions 3 Restricting the domain of trig functions

We can make the sine function one-to-one if we restrict the domain to a This new restricted sine function is one-to-one; it satisfies the horizontal region (of length π) in which each output occurs exactly once. line test! (Here it is drawn again, with the x-axis stretched out to make it easier to If we restrict the domain of sine to [− π , π ] then suddenly the previous 2 2 see.) graph looks like this.

Smith (SHSU) Elementary Functions 2013 7 / 27 Smith (SHSU) Elementary Functions 2013 8 / 27 Restricting the domain of trig functions The arcsine and arcosine functions

Now we are ready to create an inverse function. The restricted sine π π function has domain [− 2 , 2 ] and range [−1, 1]. The inverse function for sin(x) is called “inverse sine” or “arc sine.” (I will If we swap the inputs and outputs we have a new function with domain π π try to use the term “arcsine”, written arcsin(x).) The arcsine function has [−1, 1] and range [− 2 , 2 ]. π π domain [−1, 1] and range [− 2 , 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 between −1 and 1, the arcsine function takes in a value between −1 and 1 and gives out a corresponding angle.

1 π π 1 For example, arcsin( 2 ) must be the angle 6 since sin( 6 ) = 2 .

Smith (SHSU) Elementary Functions 2013 9 / 27 Smith (SHSU) Elementary Functions 2013 10 / 27 The inverse of a trig function

Elementary Functions In the next presentation, we will look in depth at the inverse functions of Part 4, Trigonometry the other trig functions. Lecture 4.6b, Inverse functions for cosine, tangent and secant

(End) Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 11 / 27 Smith (SHSU) Elementary Functions 2013 12 / 27 The Arccosine function The Arccosine function

In the previous presentation we created an inverse function for sin x by For cosine we will instead choose the restricted domain [0, π] so that each restricting the domain of sin x to the interval [− π , π ]. 2 2 output occurs exactly once. 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 π [− 2 , 2 ]. Also cosine is nonnegative on this interval and we want to choose a domain that represents all of the range [−1, 1].

Smith (SHSU) Elementary Functions 2013 13 / 27 Smith (SHSU) Elementary Functions 2013 14 / 27 The Arccosine function The other inverse trig functions

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. 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 .

We choose those quadrants with the following properties: π 1 We always include the first quadrant ([0, 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 of the domain is Quadrant II for cosine and secant and Quadrant IV for sine and cosecant.

Smith (SHSU) Elementary Functions 2013 15 / 27 Smith (SHSU) Elementary Functions 2013 16 / 27 The Arctangent function The Arctangent function

The tangent function, like all trig functions, is periodic. It has period π. 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 π [− 2 , 2 ]. (For the tangent function we will include negative angles in Quadrant IV π π so that we don’t cross a place (such as 2 or − 2 ) where the tangent is undefined.)

Smith (SHSU) Elementary Functions 2013 17 / 27 Smith (SHSU) Elementary Functions 2013 18 / 27 The Arctangent function The Arctangent function

In the figure below, we hide (in light yellow) the other branches of the Now we are ready to create the inverse function. π π tangent function and focus on the interval [− 2 , 2 ] where the tangent function is one-to-one.

Smith (SHSU) Elementary Functions 2013 19 / 27 Smith (SHSU) Elementary Functions 2013 20 / 27 Arcsecant and Arccosecant The inverse of a trig function

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 In the next presentation, we will work through some problems using inverse domain as its reciprocal, sin(x). trig functions. 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 (End) [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.)

Smith (SHSU) Elementary Functions 2013 21 / 27 Smith (SHSU) Elementary Functions 2013 22 / 27 Applied Problems with Inverse Trig Functions

A guy wire 1000 feet long is attached to the top of a tower. When pulled Elementary Functions taut it touches level ground 360 feet from the base of the tower. What Part 4, Trigonometry angle does the wire make with the ground? Lecture 4.6c, Applied Problems with Inverse Trig Functions Solution. The wire forms the of a right in which the right angle Dr. Ken W. Smith 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 Sam Houston State University 360 triangle with the angle θ between those two sides. So cos θ = 1000 = 0.36. ◦ 2013 Therefore θ = arccos(0.36) ≈ 1.20253 ≈ 68.9 .

Smith (SHSU) Elementary Functions 2013 23 / 27 Smith (SHSU) Elementary Functions 2013 24 / 27 Applied Problems with Inverse Trig Functions Applied Problems with Inverse Trig Functions

Let’s do one more problem similar to one found on WebAssign. My back yard needs watering. I up a sprinkler to water all the dry Find all angles which satisfy the sin θ = 0.2. 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 Solution. the sprinkler goes back and forth along the fence, covering all 100 feet? 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 Solution. The sprinkler needs to rotate so that it moves 50 feet in both second quadrant, with reference angle equal to arcsin(0.2). The angle in directions from the center of the fence. The sprinkler, the center of the the second quadrant is fence and one end of the fence form a (with right angle at π − arcsin(0.2) ≈ 2.9402 ≈ 180◦ − 11.54◦ = 168.46◦. the center of the fence) with short sides of lengths 35 feet and 50 feet. Let 50 θ = arctan( ) ≈ 0.96 ≈ 55◦. But there are an infinite number of solutions to the equation sin θ = 0.2. 35 Since the sine function is periodic with period 2π, take any solution and This is the angle the sprinkler must cover from the center of the fence to add 2π to it to get another solution! So if k is an , one end. So 2θ ≈ 1.92 radians ≈ 110◦ is the total angle the sprinkler arcsin(0.2) + 2πk is a solution as is (π − arcsin(0.2)) + 2πk. must rotate since we want the sprinkler to cover the fence from one end to In set notation, our solution set for the equation sin θ = 0.2 is the other. {arcsin(0.2) + 2πk : k ∈ Z} ∪ {π − arcsin(0.2) + 2πk : k ∈ Z}. Smith (SHSU) Elementary Functions 2013 25 / 27 Smith (SHSU) Elementary Functions 2013 26 / 27 Inverse Trig Functions

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

(End)

Smith (SHSU) Elementary Functions 2013 27 / 27