Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 356 Convert from DMS to decimal form. 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 356 Convert from decimal form to DMS. 5.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 356 Convert from decimal or DMS to radians. 9.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Homework, Page 356 Convert from decimal or DMS to radians. 13.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 356 Convert from radians to degrees. 17.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 356 Convert from radians to degrees. 21.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 356 Use the appropriate arc length formula to find the missing information. 25.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 356 Use the appropriate arc length formula to find the missing information. 29.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 356 A central angle θ intercepts arcs s 1 and s 2 on two concentric circles with radii r 1 and r 2, respectively. Find the missing information. 33.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3.4 inches shorter than the outside arc, what is the width of the track?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Which compass bearing is closest to a bearing of 121º?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Cathy Nyugen races on a bicycle with 13-inch radius wheels. When she is traveling at a speed of 44 ft/sec, how many revolutions per minute are her wheels making?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The captain of the tourist boat Julia follows a 038º course for 2 miles and then changes course to 047º for the next 4 miles. Draw a sketch of this trip.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page A simple pulley with the given radius r used to lift heavy objects is positioned 10 feet above ground level. Given the pulley rotates θº, determine the height to which the object is lifted. a. r = 4 in, θ = 720º b. r = 2 ft, θ = 180º
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page If horse A is twice as far as horse B from the center of a merry-go-round, then horse A travels twice as fast as horse B. Justify your answer. True, since all points on a given radius have the same angular displacement, horse A will travel twice as far as horse B for the same angular displacement, thereby traveling twice as fast.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page A bicycle with 26-inch diameter wheels is traveling at 10 mph. To the nearest whole number, how many revolutions does each wheel make per minute? a. 54 b. 129 c. 259 d. 406 e. 646
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 356 Find the difference in longitude between the given cities. 65. Minneapolis and Chicago
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 356 Assume the cities have the same longitude and find the distance between them in nautical miles. 69. New Orleans and Minneapolis
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Control tower A is 60 miles east of control tower B. At a certain time, an airplane bears 340º from control tower A and 037º from control tower B. Use a drawing to model the exact location of the airplane.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.2 Trigonometric Functions of Acute Angles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review 1. Solve for x. x Solve for x. 6 3 x
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 1. Solve for x. x Solve for x. 6 3 x
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 Right Triangle Trigonometry Two Famous Triangles Evaluating Trigonometric Functions with a Calculator Applications of Right Triangle Trigonometry … and why The many applications of right triangle trigonometry gave the subject its name.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leading Questions The functions y = secant x and y = cosecant x are reciprocal functions. Given the values of two primary trig functions, we can calculate the values of the others. Our left hand provides a key to the basic trig functions that is always with us. Given one angle and one side of a right triangle, we can find the other angle and sides. Slide 4- 26
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Position An acute angle θ in standard position, with one ray along the positive x-axis and the other extending into the first quadrant.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Trigonometric Functions of 45 º Find the values of all six trigonometric functions for an angle of 45 º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Trigonometric Functions of 60 º Find the values of all six trigonometric functions for an angle of 60º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Trigonometric for General Triangles Find the values of all six trigonometric functions for the angle x in the triangle shown.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Trigonometric Functions of Five Common Angles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trig Value Memory Aid Slide 4- 33
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Common Calculator Errors When Evaluating Trig Functions Using the calculator in the wrong angle mode (degree/radians) Using the inverse trig keys to evaluate cot, sec, and csc Using function shorthand that the calculator does not recognize Not closing parentheses
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Trigonometric for General Triangles Find the exact value of the sine of 60º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Right Triangle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Word Problem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Following Questions The circular functions get their name from the fact that we go around in circles trying to understand them. Angles are commonly measured counterclock- wise from the initial side to the terminal side. Periods of functions are concerned with the frequency of their repetition. A unit circle has a diameter of one and is located wherever convenient. Slide 4- 38
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Review Section 4.2 Page 366, Exercises: 1 – 73 (EOO)
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 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 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 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 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