Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 1 Trigonometric Functions

Copyright © 2013, 2009, 2005 Pearson Education, Inc Angles 1.2 Angle Relationships and Similar Triangles 1.3 Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions 1

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Angles 1.1 Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 For an angle measuring 55°, find the measure of its complement and its supplement. 1.1 Example 1 Finding the Complement and the Supplement of an Angle (page 3) Complement: 90° − 55° = 35° Supplement: 180° − 55° = 125°

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Find the measure of each angle. 1.1 Example 2(a) Finding Measures of Complementary and Supplementary Angles (page 3) The two angles form a right angle, so they are complements. The measures of the two angles are and

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Find the measure of each angle. 1.1 Example 2(b) Finding Measures of Complementary and Supplementary Angles (page 3) The two angles form a straight angle, so they are supplements. The measures of the two angles are and

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Perform each calculation. 1.1 Example 3 Calculating with Degrees, Minutes, and Seconds (page 4) (a) (b)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 5) (a)Convert 105°20′32″ to decimal degrees. (b)Convert ° to degrees, minutes, and seconds.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Find the angles of least possible positive measure coterminal with each angle. 1.1 Example 5 Finding Measures of Coterminal Angles (page 6) (a)1106° (b)–150° Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°. An angle of 1106° is coterminal with an angle of 26°. An angle of –150° is coterminal with an angle of 210°.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 5 Finding Measures of Coterminal Angles (cont.) (c) –603° An angle of –603° is coterminal with an angle of 117°.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 6 Analyzing the Revolutions of a CD Player (page 7) A wheel makes 270 revolutions per minute. Through how many degrees will a point on the edge of the wheel move in 5 sec? The wheel makes 270 revolutions in one minute or revolutions per second. In five seconds, the wheel makes revolutions. Each revolution is 360°, so a point on the edge of the wheel will move

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Angles 1.2 Geometric Properties ▪ Triangles

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Find the measures of angles 1, 2, 3, and 4 in the figure, given that lines m and n are parallel. 1.2 Example 1 Finding Angle Measures (page 11) Angles 2 and 3 are interior angles on the same side of the transversal, so they are supplements.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 1 Finding Angle Measures (cont.) Angles 1 and 2 have equal measure because they are vertical angles, and angles 1 and 4 have equal measure because they are alternate exterior angles.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 The measures of two of the angles of a triangle are 33° and 26°. Find the measure of the third angle. 1.2 Example 2 Finding Angle Measures (page 12) The sum of the measures of the angles of a triangle is 360°. Let x = the measure of the third angle. The third angle measures 121°.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 In the figure, triangles DEF and GHI are similar. Find the measures of angles G and I. 1.2 Example 3 Finding Angle Measures in Similar Triangles (page 13) The triangles are similar, so the corresponding angles have the same measure.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Given that triangle MNP and triangle QSR are similar, find the lengths of the unknown sides of triangle QSR. 1.2 Example 4 Finding Side Lengths in Similar Triangles (page 14) The triangles are similar, so the lengths of the corresponding sides are proportional. PM corresponds to RQ. PN corresponds to RS. MN corresponds to QS.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 4 Finding Side Lengths in Similar Triangles (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Joey wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time that Joey, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree. 1.2 Example 5 Finding the Height of a Flagpole (page 14) Let x = the height of the tree The tree is 57 feet tall.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Trigonometric Functions 1.3 Trigonometric Functions ▪ Quadrantal Angles

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 The terminal side of an angle θ in standard position passes through the point (12, 5). Find the values of the six trigonometric functions of angle θ. 1.3 Example 1 Finding Function Values of an Angle (page 22) x = 12 and y = 5. 13

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 The terminal side of an angle θ in standard position passes through the point (8, –6). Find the values of the six trigonometric functions of angle θ. 1.3 Example 2 Finding Function Values of an Angle (page 22) x = 8 and y = –

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 2 Finding Function Values of an Angle (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Find the values of the six trigonometric functions of angle θ in standard position, if the terminal side of θ is defined by 3x – 2y = 0, x ≤ Example 3 Finding Function Values of an Angle (page 23) Since x ≤ 0, the graph of the line 3x – 2y = 0 is shown to the left of the y-axis. Find a point on the line: Let x = –2. Then A point on the line is (–2, –3).

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 3 Finding Function Values of an Angle (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Find the values of the six trigonometric functions of a 360° angle. 1.3 Example 4(a) Finding Function Values of Quadrantal Angles (page 25) The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5). 1.3 Example 4(b) Finding Function Values of Quadrantal Angles (page 25) x = 0 and y = –5 and r = 5.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Using the Definitions of the Trigonometric Functions 1.4 Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Find each function value. 1.4 Example 1 Using the Reciprocal Identities (page 29) (a)tan θ, given that cot θ = 4. (b)sec θ, given that tan θ is the reciprocal of cot θ. sec θ is the reciprocal of cos θ.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Determine the signs of the trigonometric functions of an angle in standard position with the given measure. 1.4 Example 2 Finding Function Values of an Angle (page 30) (a)54° (b) 260° (c) –60° (a)A 54º angle in standard position lies in quadrant I, so all its trigonometric functions are positive. (b)A 260º angle in standard position lies in quadrant III, so its sine, cosine, secant, and cosecant are negative, while its tangent and cotangent are positive. (c)A –60º angle in standard position lies in quadrant IV, so cosine and secant are positive, while its sine, cosecant, tangent, and cotangent are negative.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. 1.4 Example 3 Identifying the Quadrant of an Angle (page 31) (a)tan θ > 0, csc θ < 0 (b)sin θ > 0, csc θ > 0 tan θ > 0 in quadrants I and III, while csc θ < 0 in quadrants III and IV. Both conditions are met only in quadrant III. sin θ > 0 in quadrants I and II, as is csc θ. Both conditions are met in quadrants I and II.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Decide whether each statement is possible or impossible. 1.4 Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function (page 32) (a) cot θ = –0.999 (b)cos θ = –1.7 (c)csc θ = 0 (a) cot θ = –0.999 is possible because the range of cot θ is (b) cos θ = –1.7 is impossible because the range of cos θ is [–1, 1]. (c) csc θ = 0 is impossible because the range of csc θ is

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 Angle θ lies in quadrant III, and Find the values of the other five trigonometric functions. 1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (page 32) Since and θ lies in quadrant III, then x = –5 and y = –8.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 5 Finding All Function Values Given One Value and the Quadrant (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 6 Finding Other Function Values Given One Value and the Quadrant (page 34) Find cos θ and tan θ given that sin θ and cos θ > 0. Reject the negative root.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 6 Finding Other Function Values Given One Value and the Quadrant (cont.)

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 7 Using Identities to Find Function Values (page 35) Find sin θ and cos θ given that cot θ and θ is in quadrant II. Since θ is in quadrant II, sin θ > 0 and cos θ < 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Example 7 Finding Other Function Values Given One Value and the Quadrant (cont.)