Transcript Slide 4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Chapter 4 Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.1 Angles and Their Measures Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review 1. Find the circumference of the circle with a radius of 4.5 in. 2. Find the radius of the circle with a circumference of 14 cm. 3. Given s r . Find s if r 2.2 cm and 4 radians. 4. Convert 65 miles per hour into feet per second. 5. Convert 9.8 feet per second to miles per hour. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Quick Review Solutions 1. Find the circumference of the circle with a radius of 4.5 in. 9 in 2. Find the radius of the circle with a circumference of 14 cm. 7 / cm 3. Given s r . Find s if r 2.2 cm and 4 radians. 8.8 cm 4. Convert 65 miles per hour into feet per second. 95.3 feet per second 5. Convert 9.8 feet per second to miles per hour. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.681 miles per hour Slide 4- 5 What you’ll learn about The Problem of Angular Measure Degrees and Radians Circular Arc Length Angular and Linear Motion … and why Angles are the domain elements of the trigonometric functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Leading Questions Angles may be measured in degrees or radials. 2π radians = 360º There are 45 minutes in a degree. There are 60 nautical miles in a degree of latitude when measured at the equator or a degree of longitude measured anywhere. Angular measurements in degrees, minutes, and seconds are used by surveyors. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Why 360º? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Degree Defined If a straight angle is divided into 180 equal parts, each of the parts equals one degree. Degrees may be expressed in decimal form. Or less commonly, in degrees, minutes, and seconds (referred to as DMS) Each degree is divided into 60 equal minutes and each minute is divided into 60 equal seconds which, in turn, may be expressed in decimal units Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Example Converting Between Decimal and DMS Measurements Convert 36.359º into DMS units Convert 45º 37’ 46” into decimal units Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Navigation In navigation, the course or bearing of an object is usually given as the angle of the line of sight measured clockwise from due north. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Radian A central angle of a circle has a measure of 1 radian if it intercepts an arc with the same length as the radius. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Degree-Radian Conversion 180 To convert radians to degrees, multiply by . radians radians To convert degrees to radians, multiply by . 180 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Example Working with Radian Measure How many radians are in 60 degrees? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Arc Length Formula (Radian Measure) If is a central angle in a circle of radius r , and if is measured in radians, then the length s of the intercepted arc is given by s r . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Arc Length Formula (Degree Measure) If is a central angle in a circle of radius r , and if is measured in degrees, then the length s of the intercepted r arc is given by s . 180 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Example Perimeter of a Pizza Slice Find the perimeter of a 30 slice of a large 8 in. radius pizza. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Angular and Linear Motion Angular speed is measured in units like revolutions per minute. Linear speed is measured in units like miles per hour. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Example Converting Rotational Speed to Linear Speed How fast is a car traveling in miles per hour if its tires are rotating at 850 rpm and the tire diameter is 28.63 inches? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 Nautical Mile A nautical mile (naut mi or nm) is the length of 1 minute of arc along Earth’s equator. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 Distance Conversions 1 statute mile 0.87 nautical miles 1 nautical mile 1.15 statute miles Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 Following Questions The basic trigonometric functions are: sine, cosine and cosecant. Calculators can only find the values of trig functions for degrees. If we know one acute angle and one side in a right triangle, we can determine the other angles and sides. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Homework Review Section: 4.1 Page 356, Exercises: 1 – 73 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 4.2 Trigonometric Functions of Acute Angles Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review 1. Solve for x. x 2 3 2. Solve for x. 6 x 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 Quick Review 3. Convert 9.3 inches to feet. a 4. Solve for a. 0.45 20 36 5. Solve for b. 1.72 b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 26 Quick Review Solutions 1. Solve for x. x 2 x 13 3 2. Solve for x. 6 x x3 3 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Quick Review Solutions 3. Convert 9.3 inches to feet. 0.775 feet a 4. Solve for a. 0.45 9 20 36 5. Solve for b. 1.72 900 / 43 20.93 b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 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 Slide 4- 29 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 4- 30 Trigonometric Functions Let be an acute angle in the right ABC. Then opp hyp sine sin cosecant csc hyp opp adj hyp cosine cos secant sec hyp adj opp adj tangent tan cotangent cot adj opp Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 31 Example Evaluating Trigonometric Functions of 45º Find the values of all six trigonometric functions for an angle of 45º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 32 Example Evaluating Trigonometric Functions of 60º Find the values of all six trigonometric functions for an angle of 60º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 Example Evaluating Trigonometric for General Triangles Find the values of all six trigonometric functions for the triangle shown. 5 7 a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley x Slide 4- 34 Trigonometric Functions of Five Common Angles sin cos 0 0 2 1 2 2 2 3 2 4 2 4 2 3 2 2 2 1 2 0 2 30 45 60 90 tan x sin cos tan x 0 0 0 0 1 0 0 30 1 2 3 2 3 3 45 2 2 3 2 2 2 1 2 3 3 1 3 D.N .E. 6 4 60 3 90 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 0 6 1 4 3 3 D.N .E. 2 Slide 4- 35 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 4- 36 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 4- 37 Example Solving a Right Triangle A right triangle with a hypotenuse of 5 inches includes a 43 angle. Find the measures of the other two angles and the lengths of the other two sides. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 38 Example Solving a Word Problem Karen places her surveyor's telescope on the top of a tripod five feet above the ground. She measures an 8 elevation above the horizontal to the top of a tree that is 120 feet away. How tall is the tree? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39