Salundagui
pangit you
CHAPTER 6: INVERSE CIRCULAR FUNCTIONS AND TRIGONOMETRIC EQUATIONS
1. INVERSE CIRCULAR FUNCTIONS
• Horizontal Line Test
o Any horizontal line will intersect the graph of a one-to-one function in at most one point
• Inverse Function
o The inverse function of the one-to-one function [pic] is defined as [pic]
• Summary of Inverse Functions
1. In a one-to-one function, each x-value corresponds to only one y-value and each y-value corresponds to only one x-value.
2. If a function [pic] is one-to-one, then [pic] has an inverse function [pic]
3. The domain of [pic] is the range of [pic] and the range of [pic] is the domain of [pic]
4. The graphs of [pic] and [pic] are reflections of each other across the line [pic]
5. To find [pic] from [pic] follow these steps:
Step 1 Replace [pic] with [pic] and interchange [pic] and [pic] Step 2 Solve for [pic] Step 3 Replace [pic] with [pic]
• Inverse Sine Function
o Since the graph of [pic] is not one-to-one, we restrict the domain to [pic]
▪ This interval contains enough of the graph of the sine function to include all possible values of y. ▪ This interval is an accepted convention that is adopted by scientific and graphing calculators.
o The inverse circular functions are used in calculus to solve certain types of related rates problems and to integrate certain rational functions
|x |y=sinx |(x,y) |
|[pic] |-1 |[pic] |
|[pic] |[pic] |[pic] |
|0 |0 |[pic] |
|[pic] |[pic] |[pic] |
|[pic] |1 |[pic] |
[pic] • The graph is continuous over the entire
CHAPTER 6: INVERSE CIRCULAR FUNCTIONS AND TRIGONOMETRIC EQUATIONS
1. INVERSE CIRCULAR FUNCTIONS
• Horizontal Line Test
o Any horizontal line will intersect the graph of a one-to-one function in at most one point
• Inverse Function
o The inverse function of the one-to-one function [pic] is defined as [pic]
• Summary of Inverse Functions
1. In a one-to-one function, each x-value corresponds to only one y-value and each y-value corresponds to only one x-value.
2. If a function [pic] is one-to-one, then [pic] has an inverse function [pic]
3. The domain of [pic] is the range of [pic] and the range of [pic] is the domain of [pic]
4. The graphs of [pic] and [pic] are reflections of each other across the line [pic]
5. To find [pic] from [pic] follow these steps:
Step 1 Replace [pic] with [pic] and interchange [pic] and [pic] Step 2 Solve for [pic] Step 3 Replace [pic] with [pic]
• Inverse Sine Function
o Since the graph of [pic] is not one-to-one, we restrict the domain to [pic]
▪ This interval contains enough of the graph of the sine function to include all possible values of y. ▪ This interval is an accepted convention that is adopted by scientific and graphing calculators.
o The inverse circular functions are used in calculus to solve certain types of related rates problems and to integrate certain rational functions
|x |y=sinx |(x,y) |
|[pic] |-1 |[pic] |
|[pic] |[pic] |[pic] |
|0 |0 |[pic] |
|[pic] |[pic] |[pic] |
|[pic] |1 |[pic] |
[pic] • The graph is continuous over the entire