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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-1 Angles 1.1 Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles.

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Presentation on theme: "Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-1 Angles 1.1 Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles."— Presentation transcript:

1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-1 Angles 1.1 Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles

2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-2 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°

3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-3 Find the angles of least possible positive measure coterminal with each angle. 1.1 Example 2 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°.

4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Measures of Coterminal Angles (cont.) (c) –603° An angle of –603° is coterminal with an angle of 117°.

5 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-5 Perform each calculation. 1.1 Example 3 Calculating with Degrees, Minutes, and Seconds (page 4) (a) (b)

6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 4) (a)Convert 105°20′32″ to decimal degrees. (b)Convert ° to degrees, minutes, and seconds.

7 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-7 Angles 1.2 Geometric Properties ▪ Triangles

8 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-8 In the figure, triangles DEF and GHI are similar. Find the measures of angles G and I. 1.2 Example 1 Finding Angle Measures in Similar Triangles (page 14) The triangles are similar, so the corresponding angles have the same measure.

9 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-9 Given that triangle MNP and triangle QSR are similar, find the lengths of the unknown sides of triangle QSR. 1.2 Example 2 Finding Side Lengths in Similar Triangles (page 15) 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.

10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Finding Side Lengths in Similar Triangles (cont.)

11 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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. 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.

12 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Radian Measure 3.1 Radian Measure ▪ Converting Between Degrees and Radians ▪

13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved To Convert degree measure to radians. Multiply degree measure by 3.1 Converting Degrees and Radians (page 94) To Convert each radians measure to degrees. Multiply radian measure by

14 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Convert each degree measure to radians. 3.1 Example 1 Converting Degrees to Radians (page 94)

15 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Convert each radian measure to degrees. 3.1 Example 2 Converting Radians to Degrees (page 94)

16 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Functions 1.3 Trigonometric Functions ▪ Right-Triangle-Based Definitions of the Trigonometric Functions (Sec. 2.1) ▪ Quadrantal Angles

17 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the sine, cosine, and tangent values for angles D and E in the figure. 2.1 Example: Finding Trigonometric Function Values of An Acute Angle (page 46 – Cover with section 1.3)

18 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the sine, cosine, and tangent values for angles D and E in the figure. 2.1 Example: Finding Trigonometric Function Values of An Acute Angle (cont.)

19 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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

20 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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 = –

21 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Function Values of an Angle (cont.)

22 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the values of the six trigonometric functions of a 360° angle. 1.3 Example 4(a) Finding Function Values of Quadrantal Angles (page 24) The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2.

23 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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 24) x = 0 and y = –5 and r = 5.

24 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Using the Definitions of the Trigonometric Functions 1.4 Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities (skip unitl chapter 5) ▪ Quotient Identities (skip until chapter 5)

25 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

26 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

27 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

28 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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 = –.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

29 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

30 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Finding All Function Values Given One Value and the Quadrant (cont.)

31 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the five remaining trig function values for θ given sec θ = -4, given that sin θ > Extra Example Finding All Function Values Given One Value and Condition (page 37 #77) 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. The remaining functions follow

32 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Functions of Acute Angles 2.1 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

33 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Determine whether each statement is true or false. 2.1 Example 4 Comparing Function Values of Acute Angles (page 49) (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.

34 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Functions of Non-Acute Angles 2.2 Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles

35 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the reference angle for 294°. 2.2 Example 1(a) Finding Reference Angles (page 55) 294 ° lies in quadrant IV. The reference angle is 360° – 294° = 66°.

36 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the reference angle for 883°. 2.2 Example 1(b) Finding Reference Angles (page 55) Find a coterminal angle between 0° and 360° by dividing 883° by 360°. The quotient is about 2.5. The reference angle is 180° – 163° = 17°. 883° is coterminal with 163°.

37 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the values of the six trigonometric functions for 135°. 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56) The reference angle for 135° is 45°. Choose point P on the terminal side of the angle. The coordinates of P are (1, –1).

38 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 56)

39 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

40 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

41 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Evaluating an Expression with Function Values of Special Angles (page 57)

42 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Radian Measure (Part II) 3.1 Finding Function Values for Angles in Radians

43 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find each function value. 3.1 Example 3 Finding Function Values of Angles in Radian Measure (page 97) (d)

44 Copyright © 2008 Pearson Addison-Wesley. All rights reserved The Unit Circle and Circular Functions 3.3 Circular Functions ▪ Finding Values of Circular Functions ▪ Determining a Number with a Given Circular Function Value ▪

45 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the exact values of sin (–3 π ), cos (–3 π ), and tan (–3 π ). 3.3 Example 1 Finding Exact Circular Function Values (page 113) An angle of –3 π intersects the unit circle at (–1, 0).

46 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Use the figure to find the exact values of 3.3 Example 2(a) Finding Exact Circular Function Values (page 113)

47 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Use the figure and the definition of tangent to find the exact value of 3.3 Example 2(b) Finding Exact Circular Function Values (page 113)

48 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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.

49 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2(b) Finding Exact Circular Function Values (page 113) corresponds to

50 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Use reference angles and degree/radian conversion to find the exact value of 3.3 Example 2(c) Finding Exact Circular Function Values (page 113) In standard position, 330° lies in quadrant IV with a reference angle of 30°, so

51 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Approximate the value of s in the interval if 3.3 Example 4(b) Finding a Number Given Its Circular Function Value (page 114) Recall that and in quadrant IV, tan s is negative.

52 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Applications of Radian Measure 3.2 Arc Length on a Circle ▪ Area of a Sector of a Circle

53 Copyright © 2008 Pearson Addison-Wesley. All rights reserved A circle has radius cm. Find the length of the arc intercepted by a central angle having each of the following measures. 3.2 Example 1 Finding Arc Length Using s = rθ (page 101) Convert θ to radians.

54 Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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? 3.2 Example 4 Finding an Angle Measure Using s = rθ (page 102) 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.

55 Copyright © 2008 Pearson Addison-Wesley. All rights reserved The arc length on the smaller gear is 3.2 Example 4 Finding an Angle Measure Using s = rθ (cont.) 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°.

56 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the area of a sector of a circle having radius ft and central angle 108.0°. 3.2 Example 5 Finding the Area of a Sector (page 103) The area of the sector is about sq ft.

57 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Linear and Angular Speed 3.4 Linear Speed ▪ Angular Speed

58 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Suppose that P is on a circle with radius 15 in., and ray OP is rotating with angular speed radian per second. 3.4 Example 1 Using Linear and Angular Speed Formulas (page 122) (a)Find the angle generated by P in 10 seconds.

59 Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Exact Circular Function Values (cont.) (b)Find the distance traveled by P along the circle in 10 seconds. (c)Find the linear speed of P in inches per second. from part (a) from part (b)


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