Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 366 Find the values of all six trigonometric functions of the angle x. 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 366 Find the values of all six trigonometric functions of the angle x. 5.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 9.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 13.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 366 Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions. 17.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 366 Evaluate without using a calculator. 21.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 366 Evaluate using a calculator. Given an exact value. 25.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 29.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 33.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Evaluate using a calculator, giving answers to three decimal places. 37.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians. 41.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians. 45.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Solve for the variable shown. 49.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Solve for the variable shown. 53.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 366 Solve for the variable shown. 57.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page A guy wire from the top of a radio tower forms a 75º angle with the ground at a 55 ft distance from the foot of the tower. How tall is the tower?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page A surveyor wanted to measure the length of a lake. Two assistants, A and C, positioned themselves at opposite ends of the lake and the surveyor positioned himself 100 feet perpendicular to the line between the assistants and on the perpendicular line from the assistant C. If the angle between his lines of sight to the two assistants is 79º12‘42“, what is the length of the lake?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Which of the following expressions does not represent a real number? a.sin 30º b. tan 45º c. cod 90º d. csc 90º e. sec 90º
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The table is a simplified trig table. Which column is the values for the sine, the cosine, and the tangent functions? The second column is tangent values, because tangent can be greater than one, the third is sine values, because they are increasing and the fourth column is cosine values because they are decreasing. Angle??? 40º º º º º º
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.3 Trigonometry Extended: The Circular Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16-point unit circle … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leading Questions We may substitute any real number n for θ in any trig function and find the value of the function. Cosine is negative in the fourth quadrant. Coterminal angles have the same measure. Quadrantal angles have their terminal sides in the center of the quadrants. The period of a trig function tells us how often it takes on identical values. Slide 4- 22
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Initial Side, Terminal Side
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Positive Angle, Negative Angle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Coterminal Angles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Coterminal Angles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Trig Functions Determined by a Point in Quadrant I
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Trigonometric Functions of any Angle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Evaluating Trig Functions of a Nonquadrantal Angle θ 1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. 2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. 3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. 4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. 5. Use the coordinates of point P and the definitions to determine the six trig functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Signs of Trigonometric Functions Slide 4- 31
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Reference Angles The acute angle made by the terminal side of an angle and the x-axis is called the reference angle. The absolute value of each trig function is equal to the absolute value of the same trig function of the reference angle in the first quadrant. The sign of the trig function is determined by the quadrant in which the terminal side lies. Slide 4- 32
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating More Trig Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using one Trig Ratio to Find the Others
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Unit Circle The unit circle is a circle of radius 1 centered at the origin.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Trigonometric Functions of Real Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Periodic Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The 16-Point Unit Circle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Following Questions Graphs of the sine function may be stretched vertically, but not horizontally. Horizontal stretches of the cosine function are the result of changes in its period. Horizontal translations of the sine function are the result of phase shifts. Sinusoids are functions whose graphs have the shape of the sine curve. Sinusoids may be used to model periodic behavior. Slide 4- 39
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Homework Assignment #28 Review Section 4.3 Page 381, Exercises: 1 – 69 (EOO) Quiz next time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.4 Graphs of Sine and Cosine: Sinusoids
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about The Basic Waves Revisited Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sinusoid
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Amplitude of a Sinusoid
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Amplitude Find the amplitude of each function and use the language of transformations to describe how the graphs are related. (a) (b) (c)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Period of a Sinusoid
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Period and Frequency Find the period and frequency of each function and use the language of transformations to describe how the graphs are related. (a) (b) (c)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Horizontal Stretch or Shrink and Period
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Frequency of a Sinusoid
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Combining a Phase Shift with a Period Change
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Graphs of Sinusoids
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Constructing a Sinusoidal Model using Time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Constructing a Sinusoidal Model using Time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Constructing a Sinusoidal Model
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Constructing a Sinusoidal Model