Haberman MTH Section I: The Trigonometric Functions Chapter 6: Inverse Trig Functions As we studied in MTH, the inverse of a function reverses the roles of the inputs and the outputs (For more information on inverse functions, check out these MTH lecture notes) For example, if f and f are inverses of one another and if f ( a) b, then f ( b) a Inverse functions are extremel valuable since the undo one another and allow us to solve equations For example, we can solve the equation x 0 b using the inverse of the cubing function, namel the cube-root function, to undo the cubing involved in the equation: x 0 x 0 x 0 As we studied in MTH, the cubing function has an inverse function because each output value corresponds to exactl one input value (eg, the onl number whose cube is 8 is ) This means that the cubing function is one-to-one, and it s onl one-to-one functions whose inverses are also functions Unfortunatel, the trig functions aren t one-to-one so, in their natural form, the don t have inverse functions For example, consider the output for the sine function: this output corresponds to the inputs,,, 7, etc; see Figure 6 6 6 6 that represent some of the locations where the sine function reaches Figure : The graph of sin( t) with red dots on the line the output In order to be a one-to-one function that has an inverse, the graph can onl reach each output value once Since inverse functions can be so valuable, we reall want inverse trig functions, so we need to restrict the domains of the functions to intervals on which the are one-to-one, and then we can construct inverse functions Let s start b constructing the inverse of the sine function
Haberman MTH Section I: Chapter 6 In order to construct the inverse of the sine function, we need to restrict the domain to an interval on which the function is one-to-one, and we need to choose an interval of the domain that utilizes the entire range of the sine function, [, ] Following tradition, we well choose the interval, In Figure, this interval of the sine function is highlighted; notice that this on this interval, the function is one-to-one and has the same range as the sine function Figure : The interval, of the graph of sin( ) t ; on this interval, the sine function is one-to-one and has the same range as the entire sine function Recall that when we construct the inverse of a function we need to reverse the rolls of the inputs and the outputs, so that the inputs for the original function become the outputs for the inverse function, and the outputs for the original become the inputs for the inverse DEFINITION: The inverse sine function, denoted following: sin ( t), is defined b the If and sin( ) t, then sin ( t) B construction, the range of sin ( t) is,, and the domain is the same as the range of the sine function: [, ] Note that the inverse sine function is often called the arcsine function and denoted arcsin( t) Now let s construct the inverse of the cosine function Like the sine function, the cosine function isn t one-to-one so we ll need to restrict its domain to construct the inverse cosine function As we can see in Figure, the cosine function is one-to-one on the interval [0, ] and, on this interval, the graph utilizes the entire range of the cosine function, [, ] So we can define the inverse cosine function on this portion of the cosine function
Haberman MTH Section I: Chapter 6 Figure : The interval 0, of the graph of cos( t) ; on this interval, the cosine function is one-to-one and has the same range as the entire cosine function Recall that when we construct the inverse of a function we need to reverse the rolls of the inputs and the outputs, so that the inputs for the original function become the outputs for the inverse function, and the outputs for the original become the inputs for the inverse DEFINITION: The inverse cosine function, denoted the following: cos ( t), is defined b If 0 and cos( ) t, then cos ( t) B construction, the range of cos ( t) is [0, ], and the domain is the same as the range of the cosine function: [, ] Note that the inverse cosine function is often called the arccosine function and denoted arccos( t) Ke Point: As we ve discussed in Part of Chapter, we can denote powers of trigonometric functions b putting the exponent between the function sin( t) sin ( t) The name and the input variable; for example, definition above implies that inverse function notation looks like the sine function raised to the power (ie, the reciprocal of the sine function), but the reciprocal of a function isn t the same as its inverse! In order to avoid ambiguous notation, the notation sin ( t) alwas refers to the inverse function If ou want to denote the reciprocal of the sine function, ou need to use the notation sin( t) : t sin( ) csc( t) sin( t) but csc( t) sin ( x)!
Haberman MTH Section I: Chapter 6 Now let s define the inverse tangent function Recall that the tangent function is one-to-one on the interval, ; since the period of tangent is units, this interval represents a complete period of tangent In order to construct the inverse tangent function, we restrict the tangent function to the interval, DEFINITION: The inverse tangent function, denoted the following: If and tan( ) B construction, the range of t, then tan ( t), is defined b tan ( t) tan ( t) is,, and the domain is the same as the range of the tangent function: Note that the inverse tangent function is often called the arctangent function and denoted arctan( t) EXAMPLE : a Evaluate sin b Evaluate c Evaluate cos (0) tan () SOLUTION: a To evaluate sin( p) sin, we need to find a value, p, such that p and Our experience tells us that p 6 Thus, sin 6 b To evaluate cos (0) cos( p) 0 Our experience tells us that, we need to find a value, p, such that 0 p and p Thus, cos (0) c To evaluate tan (), we need to find a value, p, such that tan( p) Our experience tells us that p Thus, tan () p and
Haberman MTH Section I: Chapter 6 EXAMPLE : a Evaluate sin sin SOLUTION: b Evaluate cos cos c Evaluate tan tan a To evaluate sin sin, we need to first evaluate find sin find a value, p, such that p and p Thus, sin b To evaluate cos cos sin( p) Now we can evaluate sin sin sin sin sin, so we need to Our experience tells us that :, we need to first evaluate find cos, so we need to find a value, p, such that 0 p and cos( p) Our experience tells us that p Thus, cos Now we can evaluate cos cos cos cs o cos : c To evaluate tan tan, we need to first evaluate find tan to find a value, p, such that us that p p, so we need and tan( p) Our experience tells Thus, tan Now we can evaluate tan tan tan tan tan :
Haberman MTH Section I: Chapter 6 6 Notice that the answers to all three parts of Example are exactl what we should have expected the answers to be since inverse functions undo each other But we have to be careful since the inverse sine, cosine, and tangent functions are NOT the inverses of the complete sine, cosine, and tangent functions The next example should help explain wh we need to be careful with the inverse trigonometric functions EXAMPLE : a Evaluate sin sin SOLUTION: b Evaluate cos cos( ) c Evaluate tan tan a Based on what we noticed in the last example and what we know about how inverse functions undo each other, we might assume that sin sin is equal to but this isn t true (It can t possibl be true since the answer to this question is an output for the inverse sine function and isn t in the range of this function,, ) So sin sin sin sin sin sin since since sin and is equal to, not, since isn t in the range of sin ( t) b Since inverse functions undo each other, we might assume that cos cos equal to, but this is NOT true (Notice that it can t possibl be true since the answer to this question is an output for the inverse cosine function and isn t in the range of this function, 0, ) cos cos co cos since cos 0 since s 0 and 0 0 is So cos cos is equal to 0, not, since isn t in the range of cos ( t)
Haberman MTH Section I: Chapter 6 7 c Since inverse functions undo each other, we might assume that tan tan equal to, but this is NOT true (Notice that it can t possibl be true since the answer to this question is an output for the inverse tangent function and isn t in the range of this function,, ) So tan tan tan n tan tan tan ta since since and is equal to, not, since isn t in the range of tan ( t) is