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1.1 Angles Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-1 1.1 Example 1 Finding the Complement and the Supplement of an Angle (page 3) For an angle measuring 55°, find the measure of its complement and its supplement. Complement: 90° − 55° = 35° Supplement: 180° − 55° = 125° Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-2 1.1 Example 2 Finding Measures of Coterminal Angles (page 6) Find the angles of least possible positive measure coterminal with each angle. (a) 1106° 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°. (b) –150° An angle of –150° is coterminal with an angle of 210°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-3 1.1 Example 2 Finding Measures of Coterminal Angles (cont.) (c) –603° An angle of –603° is coterminal with an angle of 117°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-4 1.1 Example 3 Calculating with Degrees, Minutes, and Seconds (page 4) Perform each calculation. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-5 1.1 Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 4) (a) Convert 105°20′32″ to decimal degrees. (b) Convert 85.263° to degrees, minutes, and seconds. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-6 1.2 Angles Geometric Properties ▪ Triangles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-7 1.2 Example 1 Finding Angle Measures in Similar Triangles (page 14) In the figure, triangles DEF and GHI are similar. Find the measures of angles G and I. The triangles are similar, so the corresponding angles have the same measure. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-8 1.2 Example 2 Finding Side Lengths in Similar Triangles (page 15) Given that triangle MNP and triangle QSR are similar, find the lengths of the unknown sides of triangle QSR. 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 © 2008 Pearson Addison-Wesley. All rights reserved. 1-9 1.2 Example 3 Finding Side Lengths in Similar Triangles (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-10 1.2 Example 5 Finding the Height of a Flagpole (page 14) Samir 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 as Samir, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree. Let x = the height of the tree The tree is 57 feet tall. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-11 3.1 Radian Measure Radian Measure ▪ Converting Between Degrees and Radians ▪ Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-12 3.1 Converting Degrees and Radians (page 94) To Convert degree measure to radians. • Multiply degree measure by 1 8 0 To Convert each radians measure to degrees. • Multiply radian measure by Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 180 3-13 3.1 Example 1 Converting Degrees to Radians (page 94) Convert each degree measure to radians. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-14 3.1 Example 2 Converting Radians to Degrees (page 94) Convert each radian measure to degrees. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-15 1.3 Trigonometric Functions Trigonometric Functions ▪ Right-Triangle-Based Definitions of the Trigonometric Functions (Sec. 2.1) ▪ Quadrantal Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-16 2.1 Example: Finding Trigonometric Function Values of An Acute Angle (page 46 – Cover with section 1.3) Find the sine, cosine, and tangent values for angles D and E in the figure. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-17 2.1 Example: Finding Trigonometric Function Values of An Acute Angle (cont.) Find the sine, cosine, and tangent values for angles D and E in the figure. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-18 1.3 Example 1 Finding Function Values of an Angle (page 22) 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. x = 12 and y = 5. 13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-19 1.3 Example 2 Finding Function Values of an Angle (page 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. x = 8 and y = –6. 10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6 1-20 1.3 Example 2 Finding Function Values of an Angle (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-21 1.3 Example 4(a) Finding Function Values of Quadrantal Angles (page 24) Find the values of the six trigonometric functions of a 360° angle. The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-22 1.3 Example 4(b) Finding Function Values of Quadrantal Angles (page 24) Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5). x = 0 and y = –5 and r = 5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-23 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities (skip unitl chapter 5) ▪ Quotient Identities (skip until chapter 5) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-24 1.4 Example 1 Using the Reciprocal Identities (page 29) Find each function value. (a) tan θ, given that cot θ = 4. tan θ is the reciprocal of cot θ. (b) sec θ, given that sec θ is the reciprocal of cos θ. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-25 1.4 Example 2 Finding Function Values of an Angle (page 30) Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (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 © 2008 Pearson Addison-Wesley. All rights reserved. 1-26 1.4 Example 3 Identifying the Quadrant of an Angle (page 31) Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. (a) tan 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. (b) sin θ > 0, csc θ > 0 sin θ > 0 in quadrants I and II, as is csc θ. Both conditions are met in quadrants I and II. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-27 1.4 Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function (page 32) Decide whether each statement is possible or impossible. (a) cot θ = –.999 (b) cos θ = –1.7 (c) csc θ = 0 (a) cot θ = –.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 © 2008 Pearson Addison-Wesley. All rights reserved. 1-28 1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (page 32) Angle θ lies in quadrant III, and Find the values of the other five trigonometric functions. Since and y = –8. and θ lies in quadrant III, then x = –5 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-29 1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-30 1.4 Extra Example Finding All Function Values Given One Value and Condition (page 37 #77) Find the five remaining trig function values for θ given sec θ = -4, given that sin θ > 0. Since sin is positive in quadrants I & II and sec is negative in quadrants II & III we restrict our discussion to quadrant II so , r = 4 and x = - 1. 4 ( 1) y so y 2 sin 2 2 y 15 r 15 4 The remaining functions follow Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-31 2.1 Trigonometric Functions of Acute Angles Right-Triangle-Based Definitions of the Trigonometric Functions (covered with section 1.3) ▪ Cofunction Identities (skip until chapter 3) ▪ Trigonometric Function Values of Special Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-32 2.1 Example 4 Comparing Function Values of Acute Angles (page 49) Determine whether each statement is true or false. (a) tan 25° < tan 23° In the interval from 0° to 90°, as the angle increases, the tangent of the angle increases. tan 25° < tan 23° is false. (b) csc 44° < csc 40° In the interval from 0° to 90°, as the angle increases, the sine of the angle increases, so the cosecant of the angle decreases. csc 44° < csc 40° is true. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-33 2.2 Trigonometric Functions of Non-Acute Angles Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-34 2.2 Example 1(a) Finding Reference Angles (page 55) Find the reference angle for 294°. 294 ° lies in quadrant IV. The reference angle is 360° – 294° = 66°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-35 2.2 Example 1(b) Finding Reference Angles (page 55) Find the reference angle for 883°. Find a coterminal angle between 0° and 360° by dividing 883° by 360°. The quotient is about 2.5. 883° is coterminal with 163°. The reference angle is 180° – 163° = 17°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-36 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56) Find the values of the six trigonometric functions for 135°. The reference angle for 135° is 45°. Choose point P on the terminal side of the angle. The coordinates of P are (1, –1). Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-37 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-38 2.2 Example 3(a) Finding Trigonometric Function Values Using Reference Angles (page 57) Find the exact value of sin(–150°). An angle of –150° is coterminal with an angle of –150° + 360° = 210°. The reference angle is 210° – 180° = 30°. Since an angle of –150° lies in quadrant III, its sine is negative. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-39 2.2 Example 3(b) Finding Trigonometric Function Values Using Reference Angles (page 57) Find the exact value of cot(780°). An angle of 780° is coterminal with an angle of 780° – 2 ∙ 360° = 60°. The reference angle is 60°. Since an angle of 780° lies in quadrant I, its cotangent is positive. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-40 2.2 Example 4 Evaluating an Expression with Function Values of Special Angles (page 57) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-41 3.1 Radian Measure (Part II) Finding Function Values for Angles in Radians Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-42 3.1 Example 3 Finding Function Values of Angles in Radian Measure (page 97) Find each function value. 1 7 o o (d) sin sin ( 2 1 0 ) sin (1 5 0 ) 6 2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-43 3.3 The Unit Circle and Circular Functions Circular Functions ▪ Finding Values of Circular Functions ▪ Determining a Number with a Given Circular Function Value ▪ Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-44 3.3 Example 1 Finding Exact Circular Function Values (page 113) Find the exact values of sin (–3π), cos (–3π), and tan (–3π). An angle of –3π intersects the unit circle at (–1, 0). Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-45 3.3 Example 2(a) Finding Exact Circular Function Values (page 113) Use the figure to find the exact values of Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-46 3.3 Example 2(b) Finding Exact Circular Function Values (page 113) Use the figure and the definition of tangent to find the exact value of Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-47 3.3 Example 2(b) Finding Exact Circular Function Values (page 113) Moving around the unit circle units in the negative direction yields the same ending point as moving around the circle units in the positive direction. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-48 3.3 Example 2(b) Finding Exact Circular Function Values (page 113) corresponds to Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-49 3.3 Example 2(c) Finding Exact Circular Function Values (page 113) Use reference angles and degree/radian conversion to find the exact value of In standard position, 330° lies in quadrant IV with a reference angle of 30°, so Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-50 3.3 Example 4(b) Finding a Number Given Its Circular Function Value (page 114) Approximate the value of s in the interval Recall that negative. if and in quadrant IV, tan s is Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-51 3.2 Applications of Radian Measure Arc Length on a Circle ▪ Area of a Sector of a Circle Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-52 3.2 Example 1 Finding Arc Length Using s = rθ (page 101) A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having each of the following measures. Convert θ to radians. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-53 3.2 Example 4 Finding an Angle Measure Using s = rθ (page 102) Two gears are adjusted so that the smaller gear drives the larger one. If the radii of the gears are 3.6 in. and 5.4 in., and the smaller gear rotates through 150°, through how many degrees will the larger gear rotate? First find the radian measure of the angle, and then find the arc length on the smaller gear that determines the motion of the larger gear. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-54 3.2 Example 4 Finding an Angle Measure Using s = rθ (cont.) The arc length on the smaller gear is An arc with length 3π cm on the larger gear corresponds to an angle measure θ radians, where The larger gear will rotate through 100°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-55 3.2 Example 5 Finding the Area of a Sector (page 103) Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0°. The area of the sector is about 217.8 sq ft. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-56 3.4 Linear and Angular Speed Linear Speed ▪ Angular Speed Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-57 3.4 Example 1 Using Linear and Angular Speed Formulas (page 122) Suppose that P is on a circle with radius 15 in., and ray OP is rotating with angular speed radian per second. (a) Find the angle generated by P in 10 seconds. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-58 3.4 Example 1 Finding Exact Circular Function Values (cont.) (b) Find the distance traveled by P along the circle in 10 seconds. from part (a) (c) Find the linear speed of P in inches per second. from part (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-59