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The Inverse of a Trig Function

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The Inverse of a Trig Function The inverse of a trig function With many of the previous elementary functions, we are able to create Elementary Functions inverse functions. Part 4, Trigonometry 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 exponential function is a logarithm. 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 inverse function 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 sine function and tan−1 x for the then we don't know how to compute f −1(y): inverse function of tangent. 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 angle (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 horizontal line test! 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 angles.) Find (without a calculator) the exact values of the x-values such that y = f(x): following: p 2 We can fix this problem by restricting the domain of the trig functions so 1 arccos( ) 2p that the trig function is one-to-one in that specific domain. 3 2 arccos(− ) p 2 3 For example, the sine function has domain (−∞; 1) and range [−1; 1]: It 3 arcsin( ) 2 is periodic with period 2π and during each period, each output occurs 4 arctan(1) p twice. 5 arctan( p3) 6 arctan(− 3) Solutions. 1 Since arccos(x) is the inversep function of cos(px) then we seekp here an 2 π 2 2 angle θ whose cosine is 2 : Since cos( 4 ) = 2 then arccos( 2 ) should be π : p4 p 3 5π 5π 3 2 arccos(− ) = since cos( ) = − : p 2 6 6 p 2 3 π π 3 3 arcsin( 2 ) = 3 since sin( 3 ) = 2 : π π 4 arctan(1) = since tan( ) = 1: 4 p 4 p π π 5 SinceSmith (SHSU)tan( ) = 3 then arctan(Elementary Functions3) = . 2013 5 / 27 Smith (SHSU) Elementary Functions 2013 6 / 27 3 p p3 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 real number 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 unit circle. 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.
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