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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 domain, [pic] • Its x-intercepts are of the form [pic]

• Its period is [pic]

• The graph is symmetric with respect to the

origin, so the function is an odd function. For all x in the domain, [pic]

|x |[pic] or |(x,y) |

| |[pic] | |

|-1 |[pic] |[pic] |

|[pic] |[pic] |[pic] |

|0 |0 |[pic] |

|[pic] |[pic] |[pic] |

|1 |[pic] |[pic] |

[pic]

• The inverse sine function is increasing and continuous on its domain [pic] • Its x-intercept is 0, and its y-intercept is 0.

• The graph is symmetric with respect to the

origin, so the function is an odd function.

• Inverse Cosine 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 cosine function to include all possible values of y.

▪ This interval is an accepted convention that is adopted by scientific and graphing calculators.

|x |y=cosx |(x,y) |

|0 |1 |[pic] |

|[pic] |[pic] |[pic] |

|[pic] |0 |[pic] |

|[pic] |[pic] |[pic] |

|[pic] |-1 |[pic] |

[pic]

• The graph is continuous over the entire domain, [pic] • Its x-intercepts are of the form [pic]

• Its period is [pic]

• The graph is symmetric with respect to the y- axis, so the function is an even function. For all...