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Copyright © 2005 Pearson Education, Inc. Chapter 1 Trigonometric Functions Copyright © 2005 Pearson Education, Inc. 1.1 Angles Objective: Understand and apply the basic terminology of angles Warm up : Define and draw a picture of each of the following terms Line Ray Acute angle Complementary Angles Copyright © 2005 Pearson Education, Inc. Line Segment Right angle Obtuse angle Supplementary Angles Basic Terms Two distinct points determine a line called line AB. A B Line segment AB—a portion of the line between A and B, including points A and B. A B Ray AB—portion of line AB that starts at A and continues through B, and on past B. A Copyright © 2005 Pearson Education, Inc. B Slide 1-4 Basic Terms continued Angle-formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Copyright © 2005 Pearson Education, Inc. Slide 1-5 Basic Terms continued Positive angle: The rotation of the terminal side of an angle counterclockwise. Copyright © 2005 Pearson Education, Inc. Negative angle: The rotation of the terminal side is clockwise. Slide 1-6 Types of Angles The most common unit for measuring angles is the degree. Copyright © 2005 Pearson Education, Inc. Slide 1-7 Example: Complementary Angles Find the measure of each angle. Since the two angles form a right angle, they are complementary angles. Thus, k 20 k 16 90 k +20 k 16 2k 4 90 2 k 86 k 43 Copyright © 2005 Pearson Education, Inc. The two angles have measures of 43 + 20 = 63 and 43 16 = 27 Slide 1-8 Example: Supplementary Angles Find the measure of each angle. Since the two angles form a straight angle, they are supplementary angles. Thus, 6 x 7 3 x 2 180 9 x 9 180 6x + 7 3x + 2 9 x 171 x 19 Copyright © 2005 Pearson Education, Inc. These angle measures are 6(19) + 7 = 121 and 3(19) + 2 = 59 Slide 1-9 Degree, Minutes, Seconds One minute is 1/60 of a degree. 1 1' 60 or 60' 1 One second is 1/60 of a minute. 1 1 1" 60 3600 Copyright © 2005 Pearson Education, Inc. or 60" 1' Slide 1-10 Example: Calculations Perform the calculation. 27 34' 26 52' Perform the calculation. 72 15 18' 27 34' 26 52' 53 86' Write 72 as 71 60' 71 60 Since 86 = 60 + 26, the sum is written 53 15 18' 1 26' 56 42' 54 26' Copyright © 2005 Pearson Education, Inc. Slide 1-11 Example: Conversions Convert to decimal degrees. 74 12' 18" 12 18 60 3600 74 .2 .005 74 12' 18" 74 74.205 Convert to degrees, minutes, and seconds 36.624 34.624 34 .624 34 .624(60') 34 37.44' 34 37 ' .44' 34 37 ' .44(60") 34 37 ' 26.4" 34 37 ' 26.4" Copyright © 2005 Pearson Education, Inc. Slide 1-12 Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-13 Coterminal Angles A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 can be produced. Such angles are called coterminal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-14 Example: Coterminal Angles Find the angles of smallest possible positive measure coterminal with each angle. a) 1115 b) 187 Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360. o o o a) 1115 3(360 ) 35 b) 187 + 360 = 173 Copyright © 2005 Pearson Education, Inc. Slide 1-15 Homework Page 7 # 14 - 42 Copyright © 2005 Pearson Education, Inc. Slide 1-16 1.2 Objective: Compare Angle Relationships and to identify similar triangles and calculate missing sides and angles. Copyright © 2005 Pearson Education, Inc. Warm up: Use the graph at the right to find the following 1. Name a pair of vertical angles. 2. Line a and b are what kind of lines. 3. Name a pair of alternate interior angles. 1 2 a 4. Name a pair of alternate exterior angles 5. Name a pair of corresponding angles. 3 b 4 5 6 7 6. Find the measure of all the angles. Copyright © 2005 Pearson Education, Inc. 8 Angles and Relationships q m n Name Angles Rule Alternate interior angles 4 and 5 3 and 6 Angles measures are equal. Alternate exterior angles 1 and 8 2 and 7 Angle measures are equal. Interior angles on the same side of the transversal 4 and 6 3 and 5 Angle measures add to 180. Corresponding angles 2 & 6, 1 & 5, 3 & 7, 4 & 8 Angle measures are equal. Copyright © 2005 Pearson Education, Inc. Slide 1-19 Vertical Angles Vertical Angles have equal measures. Q R M N P The pair of angles NMP and RMQ are vertical angles. Copyright © 2005 Pearson Education, Inc. Slide 1-20 Parallel Lines Parallel lines are lines that lie in the same plane and do not intersect. When a line q intersects two parallel lines, q, is called a transversal. Transversal q m parallel lines n Copyright © 2005 Pearson Education, Inc. Slide 1-21 Example: Finding Angle Measures Find the measure of each marked angle, given that lines m and n are parallel. (6x + 4) (10x 80) m n The marked angles are alternate exterior angles, which are equal. Copyright © 2005 Pearson Education, Inc. 6 x 4 10 x 80 84 4 x 21 x One angle has measure 6x + 4 = 6(21) + 4 = 130 and the other has measure 10x 80 = 10(21) 80 = 130 Slide 1-22 Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180. Copyright © 2005 Pearson Education, Inc. Slide 1-23 Example: Applying the Angle Sum The measures of two of the angles of a triangle are 52 and 65. Find the measure of the third angle, x. Solution 52 65 x 180 117 x 180 x 63 65 x The third angle of the triangle measures 63. 52 Copyright © 2005 Pearson Education, Inc. Slide 1-24 Types of Triangles: Angles Copyright © 2005 Pearson Education, Inc. Slide 1-25 Types of Triangles: Sides Copyright © 2005 Pearson Education, Inc. Slide 1-26 Conditions for Similar Triangles Corresponding angles must have the same measure. Corresponding sides must be proportional. (That is, their ratios must be equal.) Copyright © 2005 Pearson Education, Inc. Slide 1-27 Example: Finding Angle Measures Triangles ABC and DEF are similar. Find the measures of angles D and E. D Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle A which = 35 A 112 35 F C 112 33 Copyright © 2005 Pearson Education, Inc. E Angle E corresponds to angle B which = 33 B Slide 1-28 Example: Finding Side Lengths Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. 32 64 16 x 32 x 1024 x 32 D A 16 112 35 64 F 32 C 112 33 48 Copyright © 2005 Pearson Education, Inc. To find side DE. B E To find side FE. 32 48 16 x 32 x 768 x 24 Slide 1-29 Example: Application A lighthouse casts a shadow 64 m long. At the same time, the shadow cast by a mailbox 3 feet high is 4 m long. Find the height of the lighthouse. The two triangles are similar, so corresponding sides are in proportion. 3 x 4 64 4 x 192 x 48 3 4 x The lighthouse is 48 m high. 64 Copyright © 2005 Pearson Education, Inc. Slide 1-30 Homework Page 14-16 # 3-13 odd, 25-35 odd, 45-56 odd Copyright © 2005 Pearson Education, Inc. Slide 1-31 1.3 Objective: To understand and apply the 6 trigonometric functions Copyright © 2005 Pearson Education, Inc. Warm up In the figure below, two similar triangles are present. Find the value of each variable. x-2y 5 74 x-5 x+y 10 74 15 Copyright © 2005 Pearson Education, Inc. Slide 1-33 Trigonometric Functions Let (x, y) be a point other the origin on the terminal side of an angle in standard position. The distance from the point to the origin is r x 2 y 2 . The six trigonometric functions of are defined as follows. y sin r r csc ( y 0) y Copyright © 2005 Pearson Education, Inc. x cos r y tan (x 0) x r sec ( x 0) x x cot (y 0) y Slide 1-34 Example: Finding Function Values The terminal side of angle in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle . r x 2 y 2 122 162 (12, 16) 16 144 256 400 20 12 Copyright © 2005 Pearson Education, Inc. Slide 1-35 Example: Finding Function Values continued x = 12 y 16 4 sin r 20 5 x 12 3 cos r 20 5 y 16 4 tan x 12 3 Copyright © 2005 Pearson Education, Inc. y = 16 r = 20 r 20 5 csc y 16 4 r 20 5 sec x 12 3 x 12 3 cot y 16 4 Slide 1-36 Example: Finding Function Values Find the six trigonometric function values of the angle in standard position, if the terminal side of is defined by x + 2y = 0, x 0. We can use any point on the terminal side of to find the trigonometric function values. Copyright © 2005 Pearson Education, Inc. Slide 1-37 Example: Finding Function Values continued Choose x = 2 x 2y 0 2 2y 0 2 y 2 y 1 The point (2, 1) lies on the terminal side, and the corresponding value of r is r 22 (1) 2 5. Copyright © 2005 Pearson Education, Inc. Use the definitions: y 1 1 5 5 sin r 5 5 5 5 x 2 2 5 2 5 r 5 5 5 5 y 1 r tan csc 5 x 2 y cos sec r 5 x 2 cot x 2 y Slide 1-38 Example: Function Values Quadrantal Angles Find the values of the six trigonometric functions for an angle of 270. First, we select any point on the terminal side of a 270 angle. We choose (0, 1). Here x = 0, y = 1 and r = 1. 1 sin 270 1 1 1 tan 270 undefined 0 1 sec 270 undefined 0 Copyright © 2005 Pearson Education, Inc. 0 cos 270 0 1 1 csc 270 1 1 0 cot 270 0 1 Slide 1-39 Undefined Function Values If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant functions are undefined. If it lies along the x-axis, then the cotangent and cosecant functions are undefined. Copyright © 2005 Pearson Education, Inc. Slide 1-40 Commonly Used Function Values sin cos tan cot sec csc 0 0 1 0 undefined 1 undefined 90 1 0 undefined 0 undefined 1 180 0 1 0 undefined 1 undefined 270 1 0 undefined 0 undefined 1 360 0 1 0 undefined 1 undefined Copyright © 2005 Pearson Education, Inc. Slide 1-41 Homework Page 25 # 18-46 even Copyright © 2005 Pearson Education, Inc. Slide 1-42 1.4 Objective: to apply the definitions of the trigonometric functions Copyright © 2005 Pearson Education, Inc. Warm-up What is the reciprocal of 2/3? 1 2/5? 0? Cos 0? Sin 0? Tan 0? Copyright © 2005 Pearson Education, Inc. Slide 1-44 Reciprocal Identities 1 sin csc 1 cos sec 1 tan cot 1 csc sin 1 sec cos 1 cot tan Copyright © 2005 Pearson Education, Inc. Slide 1-45 Example: Find each function value. 2 cos if sec = 3 Since cos is the reciprocal of sec 1 1 3 cos 2 sec 3 2 15 sin if csc 3 1 3 sin 15 15 3 3 • 15 3 • 15 15 15 • 15 Copyright © 2005 Pearson Education, Inc. 15 5 Slide 1-46 Signs of Function Values in sin cos tan cot sec csc Quadrant I + + + + + + II + + III + + IV + + Copyright © 2005 Pearson Education, Inc. Slide 1-47 Example: Identify Quadrant Identify the quadrant (or quadrants) of any angle that satisfies tan > 0, cot > 0. tan > 0 in quadrants I and III cot > 0 in quadrants I and III so, the answer is quadrants I and III Copyright © 2005 Pearson Education, Inc. Slide 1-48 Ranges of Trigonometric Functions For any angle for which the indicated functions exist: 1. 1 sin 1 and 1 cos 1; 2. tan and cot can equal any real number; 3. sec 1 or sec 1 and csc 1 or csc 1. (Notice that sec and csc are never between 1 and 1.) Copyright © 2005 Pearson Education, Inc. Slide 1-49 Identities Pythagorean sin 2 cos 2 1, tan 2 1 sec 2 , 1 cot 2 csc 2 Copyright © 2005 Pearson Education, Inc. Quotient sin tan cos cos cot sin Slide 1-50 Example: Other Function Values Find sin and cos if tan = 4/3 and is in quadrant III. Since is in quadrant III, sin and cos will both be negative. sin and cos must be in the interval [1, 1]. Copyright © 2005 Pearson Education, Inc. Slide 1-51 Example: Other Function Values continued We use the identity tan 2 1 sec 2 tan 2 1 sec 2 2 4 2 1 s ec 3 16 1 sec 2 9 25 sec 2 9 5 sec 3 3 cos 5 Copyright © 2005 Pearson Education, Inc. Since sin 2 1 cos 2 , 3 sin 2 1 5 9 sin 2 1 25 16 sin 2 25 4 sin 5 2 Slide 1-52 Homework Page 33-35 # 4-10, 16, 18, 56-62 Copyright © 2005 Pearson Education, Inc. Slide 1-53