Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 1.6 Trigonometric Functions
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Slide 1- 5 What you’ll learn about… Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions …and why Trigonometric functions can be used to model periodic behavior and applications such as musical notes.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Complete the unit circle.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 7 Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Arc Length (Radians) If a circle has radius r and an arc of that circle has measure of θ, then the length of that arc is determined by: AL = rθ Slide 1- 8
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall An angle has measure of 5π / 8 and the circle has radius of 2 units. What is the length of the arc determined by the angle? Slide 1- 9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Arc Length (Degrees) If a circle has radius r and an arc is determined by a central angle of M°, then the length of the arc is given by: Slide 1- 10
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall The radius of a circle is 6 units and the length of a particular arc of the circle is 3π / 2 units. What is the measure of the angle that determines the arc? Slide 1- 11
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Radian Measure An angle of measure θ is placed in standard position at the center of circle of radius r,
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Trigonometric Functions of θ
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Graphs of Trigonometric Functions When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable (radians) by x instead of θ.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Periodic Function, Period
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Even and Odd Trigonometric Functions The graphs of cos x and sec x are even functions because their graphs are symmetric about the y-axis. The graphs of sin x, csc x, tan x and cot x are odd functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Even and Odd Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Transformations of Trigonometric Graphs The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions. Vertical stretch or shrink Reflection about x-axis Horizontal stretch or shrink Reflection about the y-axis Horizontal shift Vertical shift
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Transformations of Trigonometric Graphs [-5, 5 ] by [-4,4]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Transformations of Trigonometric Graphs [-5, 5 ] by [-4,4]
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y = 1.5 sin 2x Specify the period, the amplitude, and identify the viewing window for the function which is graphed below. Slide 1- 24
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment part 1 page 52, # 2 – 16 evens Slide 1- 25
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Inverse Trigonometric Functions None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. The domains and ranges of the inverse trigonometric functions become part of their definitions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Inverse Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Inverse Trigonometric Functions The graphs of the six inverse trigonometric functions are shown here.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Inverse Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Solve the equation in the specified interval. Slide 1- 31
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 1.4 – 1.6
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Solve the equation in the specified interval. Slide 1- 36
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 1.4 – 1.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 1.4 – 1.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment 2 pages 53, # 26, 28, 32, 34, 38, 40, 42 Slide 1- 39