Chapter 5 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Angles and Radian Measure

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Objectives: Recognize and use the vocabulary of angles. Use degree measure. Use radian measure. Convert between degrees and radians. Draw angles in standard position. Find coterminal angles. Find the length of a circular arc. Use linear and angular speed to describe motion on a circular path.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Angles An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Angles (continued) An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Angles (continued) When we see an initial side and a terminal side in place, there are two kinds of rotations that could have generated the angle. Positive angles are generated by counterclockwise rotation. Thus, angle is positive. Negative angles are generated by clockwise rotation. Thus, angle is negative.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Angles (continued) An angle is called a quadrantal angle if its terminal side lies on the x-axis or on the y-axis. Angle is an example of a quadrantal angle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Measuring Angles Using Degrees Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360°. An acute angle measures less than 90°. A right angle measures 90°. An obtuse angle measures more than 90° but less than 180°. A straight angle measures 180°.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Measuring Angles Using Radians An angle whose vertex is at the center of the circle is called a central angle. The radian measure of any central angle of a circle is the length of the intercepted arc divided by the circle’s radius.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Radian Measure

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Computing Radian Measure A central angle, in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of ? The radian measure of is 3.5 radians.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Conversion between Degrees and Radians

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Converting from Degrees to Radians Convert each angle in degrees to radians: a. 60° b. 270° c. –300°

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Converting from Radians to Degrees Convert each angle in radians to degrees: a. b. c.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Drawing Angles in Standard Position The figure illustrates that when the terminal side makes one full revolution, it forms an angle whose radian measure is The figure shows the quadrantal angles formed by 3/4, 1/2, and 1/4 of a revolution.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Drawing Angles in Standard Position Draw and label the angle in standard position: Initial side Terminal side Vertex The angle is negative. It is obtained by rotating the terminal side clockwise. We rotate the terminal side clockwise of a revolution.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Drawing Angles in Standard Position Draw and label the angle in standard position: The angle is positive. It is obtained by rotating the terminal side counterclockwise. We rotate the terminal side counter clockwise of a revolution. Vertex Terminal side Initial side

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Drawing Angles in Standard Position Draw and label the angle in standard position: The angle is negative. It is obtained by rotating the terminal side clockwise. We rotate the terminal side clockwise of a revolution. Terminal side Initial side Vertex

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Drawing Angles in Standard Position Draw and label the angle in standard position: The angle is positive. It is obtained by rotating the terminal side counterclockwise. We rotate the terminal side counter clockwise of a revolution. Terminal side Vertex Initial side

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Degree and Radian Measures of Angles Commonly Seen in Trigonometry In the figure below, each angle is in standard position, so that the initial side lies along the positive x-axis.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Positive Angles in Terms of Revolutions of the Angle’s Terminal Side Around the Origin

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Positive Angles in Terms of Revolutions of the Angle’s Terminal Side Around the Origin (continued)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Coterminal Angles Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Finding Coterminal Angles Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following: a. a 400° angle 400° – 360° = 40° b. a –135° angle –135° + 360° = 225°

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Finding Coterminal Angles Assume the following angles are in standard position. Find a positive angle less than that is coterminal with each of the following: a. a angle b. a angle

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 The Length of a Circular Arc

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Finding the Length of a Circular Arc A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45°. Express arc length in terms of Then round your answer to two decimal places. We first convert 45° to radians:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Definitions of Linear and Angular Speed

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Linear Speed in Terms of Angular Speed

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 Example: Finding Linear Speed Long before iPods that hold thousands of songs and play them with superb audio quality, individual songs were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center. Before applying the formula we must express in terms of radians per minute:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 Example: Finding Linear Speed (continued) A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center. The angular speed of the record is radians per minute. The linear speed is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32

Chapter 5 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Right Triangle Trigonometry

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34 Objectives: Use right triangles to evaluate trigonometric functions. Find function values for Recognize and use fundamental identities. Use equal cofunctions of complements. Evaluate trigonometric functions with a calculator. Use right triangle trigonometry to solve applied problems.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35 The Six Trigonometric Functions The six trigonometric functions are: FunctionAbbreviation sine sin cosine cos tangent tan cosecant csc secant sec cotangent cot

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 36 Right Triangle Definitions of Trigonometric Functions In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 37 Right Triangle Definitions of Trigonometric Functions (continued) In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38 Example: Evaluating Trigonometric Functions Find the value of the six trigonometric functions in the figure. We begin by finding c.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 39 Function Values for Some Special Angles A right triangle with a 45°, or radian, angle is isosceles – that is, it has two sides of equal length.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 40 Function Values for Some Special Angles (continued) A right triangle that has a 30°, or radian, angle also has a 60°, or radian angle. In a triangle, the measure of the side opposite the 30° angle is one-half the measure of the hypotenuse.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 41 Example: Evaluating Trigonometric Functions of 45° Use the figure to find csc 45°, sec 45°, and cot 45°.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 42 Example: Evaluating Trigonometric Functions of 30° and 60° Use the figure to find tan 60° and tan 30°. If a radical appears in a denominator, rationalize the denominator.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 43 Trigonometric Functions of Special Angles

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 44 Fundamental Identities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 45 Example: Using Quotient and Reciprocal Identities Given and find the value of each of the four remaining trigonometric functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 46 Example: Using Quotient and Reciprocal Identities (continued) Given and find the value of each of the four remaining trigonometric functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 47 The Pythagorean Identities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 48 Example: Using a Pythagorean Identity Given that and is an acute angle, find the value of using a trigonometric identity.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 49 Trigonometric Functions and Complements Two positive angles are complements if their sum is 90° or Any pair of trigonometric functions f and g for which and are called cofunctions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 50 Cofunction Identities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 51 Using Cofunction Identities Find a cofunction with the same value as the given expression: a. b.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 52 Using a Calculator to Evaluate Trigonometric Functions To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 53 Example: Evaluating Trigonometric Functions with a Calculator Use a calculator to find the value to four decimal places: a. sin 72.8° (hint: Be sure to set the calculator to degree mode) b. csc 1.5 (hint: Be sure to set the calculator to radian mode)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 54 Applications: Angle of Elevation and Angle of Depression An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. The angle formed by the horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 55 Example: Problem Solving Using an Angle of Elevation The irregular blue shape in the figure represents a lake. The distance across the lake, a, is unknown. To find this distance, a surveyor took the measurements shown in the figure. What is the distance across the lake? The distance across the lake is approximately yards.