Warm-up – 9/18/2015 Do your warm-up in your notes 1) 2) 3) Assignment 4.7c Pg. 547 32-72 Every other even, 95, 100, 108 Agenda Warm-up 4.7 notes (day 3)- Finding values of inverse functions (domain/range) Compositions of inverses

Assignment #3A Solutions 2) 4) 6) 8) 10) 12) 14) 16) 18) 20) 22) 24) 26) 28) 30)

(4.7) Inverse Trigonometric Functions What you need to know and be able to do… Understand the graphs, domain and range of the inverse sine, cosine and tangent. Use the inverse functions to find the exact angle measure given a value of sine, cosine and tangent. Be able to find the composition of functions with their inverses and composite expressions.

Key vocabulary Domain of a function – The set of all input (x) values for which a function is defined. Range of a function – The set of all output (y) values for which a function is defined. Inverse function – A function that “reverses” another function by using the y-values as inputs and getting x-values back. Composite function – The input of one function is another function. One-to-one function - A function for which every element of the range of the function corresponds to exactly one element of the domain. “Horizontal line test” Interval – the space defined as the endpoints along the x-axis.

Review: Inverse Functions A function and its inverse function can be described as the "DO" and the "UNDO" functions.  A function takes a starting value, performs some operation on this value, and creates an output answer.  The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value. (from: http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm) Inverse functions switch domain and range from the original function.

Graph of the Sine Function Domain: All real numbers **Not one-to-one Range: - 1 to 1 This part is one-to-one if we restrict the domain. To be one-to-one, it must pass the “horizontal line test”. Therefore, we restrict the domain to accomplish this.

What is the domain and range of the Inverse Sine function? The inverse’s domain would be -1 to 1; Yet the range is not all real numbers The range must stop at and In order for this to remain a function! Domain Range

Inverse Cosine function Domain: Range:

Graph of the tangent function Domain: All real numbers except Where n is a integer Range: All real numbers

Graph of inverse tangent function Domain: All real numbers Range:

Graphs of Inverse Trigonometric Functions The basic idea of the inverse function is the same whether it is arcsin, arccos, or arctan

Finding the exact value of inverse trig functions Notation: You’ll see inverse trig functions use two equivalent notations: They both mean the same thing. The notation used in our text is . Other texts may use the other notation and Khan Academy or other video help may use In addition, we are not studying inverse cot, inverse sec or inverse csc functions.

Examples: Finding the exact value Find the exact value of the expression. A) B) C)

Examples: Using your calculator Use your calculator to find the value of the expression rounded to two decimal places. A) B)

Evaluating Compositions of Functions and their inverses For all functions and their inverses and When x in the domain of f. Inverse properties From p. 543 (do not copy for notes): The Sine Function and Its Inverse sin(sin-1 x) = x for every x in the interval [-1, 1]. sin-1(sin x) = x for every x in the interval [-/2,/2]. The Cosine Function and Its Inverse cos(cos-1 x) = x for every x in the interval [-1, 1]. cos-1(cos x) = x for every x in the interval [0, ]. The Tangent Function and Its Inverse tan(tan-1 x) = x for every real number x tan-1(tan x) = x for every x in the interval (-/2,/2).

Examples Find the exact value, if possible. Ask this question: Is the value of x in the domain of ? Is the value of x in the domain of ? If not, then we must evaluate first.

Example 1- Evaluating a Composite Trig Expression Find the exact value of

Example 4 Use a right triangle to write the expression as an algebraic expression.