Trigonometry Lecture Notes Section 6.1 Page 1 of 10

Section 6.1: Inverse Circular Functions Big Idea: An inverse function computes the input required for a function to give a desired answer. The trigonometric functions have inverses, but only for restricted intervals of their domain.

Big Skill: You should be able to state the output of the inverse trigonometric functions for key values, and use your calculator to compute the output of inverse trigonometric functions for other values.

Vertical Line test Any vertical line will intersect the graph of a function in at most one point.

Horizontal Line Test Any horizontal line will intersect the graph of a one-to-one function in at most one point.

Inverse Function The inverse function of a one-to-one function f is defined as f-1 = {( y, x) |( x , y) belongs to f }. In other words, switch the x and y values of points on the graph of a function to obtain the graph of an inverse function.

Summary of Inverse Functions  For a one-to-one function, each x-value corresponds to only one y-value, and each y- value corresponds to only one x-value (i.e., the function passes the vertical line test and the horizontal line test).  If a function f is one-to-one, then f has an inverse function, which we write as f-1.  The domain of f is the range of f-1, and the range of f is the domain of f-1.  The graphs of f and f-1 are reflections of each other across the line y = x.  To find f-1(x) from an algebraic function f(x), follow these steps: o Interchange x and y in the equation y = f(x). o Solve for y. o The resulting expression that y is equal to is f-1(x). Trigonometry Lecture Notes Section 6.1 Page 2 of 10

Practice: 1. Sow that the inverse of f( x) = x3 -1 is f-1 ( x) =3 x +1 .

2. State one way you could restrict the domain of f( x) = x2 to make it a one-to-one function.

Inverse Sine Function -1 p p y= sin ( x) or y= arcsin ( x) means that x= sin ( y) , for - 2＃y 2 .  Note that the domain of the sine function has to be restricted to make it a one-to-one function in order to define an inverse. Trigonometry Lecture Notes Section 6.1 Page 3 of 10

Practice: 3. Find y in each equation: 骣 3  y = arcsin 琪 桫2

-1 骣 1  y =sin 琪 - 桫 2

-1 骣 2  y = sin 琪 桫2

 y = sin-1 ( 2 ) Trigonometry Lecture Notes Section 6.1 Page 4 of 10

Graph of the Inverse Sine Function y= sin-1 ( x) OR y= arcsin ( x)

Domain: Range:

Table of Values:

x y= sin -1 ( x)

Notes on the graph of the inverse sine function:  The inverse sine function is increasing and continuous on its domain.  Both the x- and y- intercepts are 0.  The inverse sine function is an odd function. Trigonometry Lecture Notes Section 6.1 Page 5 of 10

Inverse Cosine Function y= cos-1 ( x) or y= arccos( x) means that x= cos( y) , for 0 ＃y p .  Note that the domain of the cosine function has to be restricted to make it a one-to-one function in order to define an inverse.

Practice: 4. Find y in each equation: 骣 3  y = arccos琪 桫2

-1 骣 1  y =cos 琪 - 桫 2

-1 骣 2  y = cos 琪 桫2

 y = cos-1 ( 2) Trigonometry Lecture Notes Section 6.1 Page 6 of 10

Graph of the Inverse Cosine Function y= cos-1 ( x) OR y= arccos( x)

Domain: Range:

Table of Values:

x y= cos-1 ( x)

Notes on the graph of the inverse cosine function:  The inverse cosine function is decreasing and continuous on its domain.  Its x-intercept is 1, and its y-intercept is /2.  The inverse cosine function is neither odd nor even. Trigonometry Lecture Notes Section 6.1 Page 7 of 10

Inverse Tangent Function -1 p p y= tan ( x) or y= arctan ( x) means that x= tan ( y) , for - 2＃y 2 .  Note that the domain of the tangent function has to be restricted to make it a one-to-one function in order to define an inverse.

Practice: 5. Find y in each equation:  y = arccos( 3)

 y = tan-1 ( 1)

 y =tan-1 ( - 1)

 y = tan-1 ( 2) Trigonometry Lecture Notes Section 6.1 Page 8 of 10

Graph of the Inverse Tangent Function y= tan-1 ( x) OR y= arctan ( x)

Domain: Range:

Table of Values:

x y= tan-1 ( x)

Notes on the graph of the inverse tangent function:  The inverse tangent function is increasing and continuous on its domain.  Both the x- and y- intercepts are 0.  The inverse tangent function is odd. p  The lines y = 2 are horizontal asymptotes. Trigonometry Lecture Notes Section 6.1 Page 9 of 10

Inverse Cotangent, Secant, and Cosecant Functions Trigonometry Lecture Notes Section 6.1 Page 10 of 10

Finding Inverse Trigonometric Functions with a Calculator

-1 - 1 骣1 -1 - 1 骣1 -1 - 1 骣1 cot(u) = tan 琪 sec(u) = cos 琪 csc(u) = sin 琪 桫u 桫u 桫u

Finding Trigonometric Functions of Inverse Trigonometric Functions (Note: there are restrictions to the domains and ranges of the formulas below that are being glossed over…) u sin sin -1 u= u -1 2 tan sin -1 (u) = ( ( )) cos( sin(u)) = 1 - u ( ) 2 1- u 2 1 -1 1 csc sin-1 u = sec sin (u) = -1 1- u ( ( )) ( ) 2 cot( sin (u)) = u 1- u u

2 -1 2 -1 1- u sin( cos(u)) = 1 - u cos( cos (u)) = u tan( cos-1 (u)) = u

-1 1 1 -1 u csc( cos (u)) = sec( cos-1 (u)) = cot( cos (u)) = 1- u2 u 1- u2

u 1 sin( tan -1 (u)) = cos( tan -1 (u)) = tan( tan-1 (u)) = u 1+ u 2 1+ u2 2 1+ u -1 2 -1 1 csc( tan-1 (u)) = sec( tan(u)) = 1 + u cot( tan (u)) = u u

-1 1 -1 u 1 sin( cot (u)) = cos( cot (u)) = tan( cot -1 (u)) = 1+ u2 1+ u2 u 2 -1 1 csc cot (u) = -1 1+ u cot cot-1 u= u ( ) 2 sec( cot (u)) = ( ( )) 1+ u u

2 u -1 -1 1 -1 2 sin( sec-1 (u)) = cos( sec (u)) = tan( sec(u)) = u - 1 u u u 1 csc( sec-1 (u)) = sec( sec-1 (u)) = u cot( sec-1 (u)) = u 2 -1 u 2 -1

2 1 -1 1 sin csc-1 u = -1 u -1 tan csc (u) = ( ( )) cos( csc (u)) = ( ) 2 u u u -1 u csc csc-1 u= u sec csc-1 (u) = -1 2 ( ( )) ( ) 2 cot( csc(u)) = u - 1 u -1