Chapter 1 and Section 2.1

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1 Chapter 1 and Section 2.1 Diana Pell Section 1.1: Angles, Degrees, and Special Triangles Angles Degree Measure Angles that measure 90 are called right angles. Angles that measure between 0 and 90 are called acute angles. 1

2 Angles that measure between 90 and 180 are called obtuse angles. If two angles have a sum of 90, then they are called complementary angles, and we say that each is the complement of the other. If two angles have a sum of 180, then they are called supplementary angles. Exercise 1. Give the complement and the supplement of each angle. a) 40 b) 110 2

3 c) θ Triangles 3

4 Pythagorean Theorem In any right triangle, the square of the length of the longest side (called the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (called legs). 4

5 Exercise 2. Solve for x in the right triangle below. 9.jpg 9.bb Figure 1: Figure 9 Exercise 3. Vertical rise of the Forest Double chair lift (Figure 10) is 1,170 feet and the length of the chair lift is 5,750 feet. To the nearest foot, find the horizontal distance covered by a person riding this lift. 10.jpg 10.bb Figure 2: Figure 10 5

6 The Triangle Exercise 4. If the shortest side of a triangle is 5, find the other two sides. Exercise 5. (You Try!) If the longest side of a triangle is 14, find the lengths of the other two sides. Exercise 6. A ladder is leaning against a wall. The top of the ladder is 4 feet above the ground and the bottom of the ladder makes an angle of 60 with the ground. How long is the ladder, and how far from the wall is the bottom of the ladder? 15.jpg 15.bb Figure 3: Figure 15 6

7 The Triangle Exercise 7. A 10-foot rope connects the top of a tent pole to the ground. If the rope makes an angle of 45 with the ground, find the length of the tent pole. 19.jpg 19.bb Figure 4: Figure 19 7

8 Section 1.2: The Rectangular Coordinate System The Distance Formula The distance between any two points (x 1, y 1 ) and (x 2, y 2 ) in a rectangular coordinate system is given by the formula r = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Exercise 8. Find the distance between the points ( 1, 5) and (2, 1). A circle is defined as the set of all points in the plane that are a fixed distance from a given fixed point. Equation of a Circle The equation of a circle with center (h, k) and radius r > 0 is given by the formula (x h) 2 + (y k) 2 = r 2 If the center is at the origin so that (h, k) = (0, 0), then the equation 8

9 becomes x 2 + y 2 = r 2 ( ) 2 2 Exercise 9. Verify that the points 2, 2 lie on a circle of radius 1 centered at the origin. and ( ) 3 2, 1 2 both Definitions 1. An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at the origin. 2. Two angles in standard position with the same terminal side are called coterminal angles. 9

10 Exercise 10. Draw 90 in standard position and find two positive angles and two negative angles that are coterminal with 90. Exercise 11. Find all angles that are coterminal with 120. Section 1.3: Trigonometric Functions 10

11 Exercise 12. Find the six trigonometric functions of θ if θ is in standard position and the point ( 2, 3) is on the terminal side of θ. Exercise 13. (You Try!) Find the six trigonometric functions of θ if θ is in standard position and the point (1, 4) is on the terminal side of θ. 11

12 Exercise 14. Find the sine and cosine of 45. Exercise 15. Find the sine and cosine of 60. Exercise 16. Find the six trigonometric functions of

13 Exercise 17. Which will be greater, tan 30 or tan 40. Algebraic Signs of Trigonometric Functions Exercise 18. If sin θ = 5 13, and θ terminates in quadrant III, find cos θ and tan θ. 13

14 Exercise 19. (You Try!) If tan θ = 4 3, and θ terminates in quadrant III, find sin θ and cos θ. Exercise 20. Find sin θ and cos θ if the terminal side of θ lies along the line y = 2x in QIII. 14

15 Section 1.4: Introduction to Identities Reciprocal Identities Exercise 21. If cos θ = 3 2, find sec θ. Exercise 22. If csc θ = 13 12, find sin θ. Ratio Identities 15

16 Exercise 23. If sin θ = 3 5 and cos θ = 4 5, find tan θ and cot θ. Exercise 24. (You Try!) If cos θ = 3 5 and sin θ = 4 5, find tan θ and cot θ. Note: sin 2 θ is a shorthand notation for (sin θ) 2. 16

17 Pythagorean Identities cos 2 θ + sin 2 θ = tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ Exercise 25. If sin θ = 3 5 and θ terminates in QII, find cos θ and tan θ. 17

18 Exercise 26. (You Try!) If sin θ = 1 3 cos θ and tan θ. and θ terminates in QIII, find 18

19 Exercise 27. If cos θ = 1 2 trigonometric ratios for θ. and θ terminates in QIV, find the remaining 19

20 Exercise 28. (You Try!) If cos θ = 5 13 the remaining trigonometric ratios for θ. and θ terminates in QII, find 20

21 Section 1.5: More on Identities Pythagorean Identities cos 2 θ + sin 2 θ = tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ Reciprocal Identities Exercise 29. Write tan θ in terms of sin θ. Exercise 30. Write sec θ tan θ in terms of sin θ and cos θ, and then simplify. 21

22 Exercise 31. (You Try!) Write tan θ csc θ in terms of sin θ and cos θ, and then simplify. Exercise 32. Add: 1 sin θ + 1 cos θ Exercise 33. (You Try!) Add: sin θ + 1 cos θ 22

23 Exercise 34. Multiply (sin θ + 2)(sin θ 5) Exercise 35. Simplify the expression x as much as possible after substituting 3 tan θ for x. Exercise 36. Prove the identity cos θ tan θ = sin θ 23

24 Exercise 37. (You Try!) Prove the identity sin θ + cos θ cot θ = csc θ Exercise 38. Prove the identity. (sin θ + cos θ) 2 = sin θ cos θ Exercise 39. (You Try!) Prove the identity. (cos θ + sin θ) 2 = 2 cos θ sin θ

25 Exercise 40. Prove the identity cos θ(sec θ cos θ) = sin 2 θ Section 2.1: Right Triangle Trigonometry 25

26 Exercise 41. Triangle ABC is a right triangle with C = 90. If a = 6 and c = 10, find the six trigonometric functions of angle A. Exercise 42. (You Try!) Triangle ABC is a right triangle with C = 90. If a = 12 and c = 15, find the six trigonometric functions of angle A. 26

27 Definition 1. Sine and cosine are cofunctions, as are tangent and cotangent, and secant and cosecant. We say sine is a cofunction of cosine, and cosine is a cofunction of sine. Definition 2. A trigonometric function of an angle is always equal to the cofunction of the complement of the angle. Exercise 43. Fill in the blanks so that each expression becomes a true statement. a) sin = cos 30 b) sec 75 = csc Two Special Triangles: and

28 Table of Exact Values for sin θ, cos θ, and tan θ Exercise 44. Simplify each expression. a) 5 sin 2 30 b) (sin 60 + cos 60 ) 2 28

29 c) tan tan 2 60 Do problems 2.1: 48, 52, 70 Homework 1.1: 17, 21, 29, 31, 35, 37, 39, 41, 43, 45, 49, 53, 57, 61, 63, 67, : 11, 13, 17, 35, 37, 41, 61, 65, 67, 71, 75, 77, : 5, 11, 13, 17, 23, 25, 29, 31, 43, 49, 51, 53, 55, 57, 61, 65, 67, 71, : 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 37, 43, : 5, 21, 27, 29, 33, 37, 39, 45, 49, 55, 61, 63, 69, 73, 79, 83, 91, : 11, 17, 19, 27, 29, 35, 39, 43, 45, 49, 51, (odd), 69, 71, 74 29