1. Make sure your calculator is in degree mode

2. review-solving right triangles

3. special right triangles

4. 30-60-90 triangles

5. I recommend setting up your sides as equations and solving each side for x.

10. Sin(xo )= Cos(x o)= Tan(xo )= O H A H O A pposite ypotenuse djacent pposite ypotenuse djacent https://www.youtube.com/watch?v=t2uPYYLH4Zo **The following are to be used in right Δs only!!**

11. EX] Write the tan, sin, and cos ratios for U. T U V 3 4 5

16. EX] Find the value of x to the nearest tenth. a.) 10 x 54

17. c.) x 1.5 28

19. EX] Solve for x to the nearest degree. x 6 8 tan x = 6 8 x = tan-1 (6/8) x ≈ 37 a.)

20. b.) 27 10 x

21. Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space aka GPS). Music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development are some of the other fields which utilize trigonometry. The Trigonometric Functions

22. 5.1 Angles and Degree Measure Goals: 1. Convert decimal degree measures to degrees, minutes, and seconds and vice versa 2. Identify angles that are co-terminal with a given angle 3. Identify reference angles. 4. Solve triangles using trig functions 5.2 Trig Ratios in Right Triangles

23. 5.1 and 5.2 Define & illustrate each term: 1. vertex 2. initial side of an angle 3. terminal side of an angle 4. standard position of an angle 5. degrees/minutes/seconds 6. quandrantal angles 7. coterminal angles 8. reference angles 9. SOH-CAH-TOA

24. Terminology vertex vertex point - a fixed end point shared by two rays forming an angle - Greek letter "theta". Used in mathematics to indicate an angle.

25. Terminology initial side - the fixed side of an angle vertex point - a fixed end point shared by two rays forming an angle initial side vertex

26. Terminology terminal side - the 2nd ray that rotates initial side - the fixed side of an angle vertex point - a fixed end point shared by two rays forming an angle initial side terminal side vertex

27. Terminology standard position - If an angle's vertex is at the origin and its initial side is along the positive x- axis terminal side - the 2nd ray that rotates initial side - the fixed side of an angle x y standard position vertex point - a fixed end point shared by two rays forming an angle vertex

28. Terminology degree - the most common unit used to measure an angle a degree is subdivided into 60 equal parts known as minutes (1') a minute is subdivided into 60 equal parts known as seconds (1") standard position - If an angle's vertex is at the origin and its initial side is along the positive x-axis terminal side - the 2nd ray that rotates initial side - the fixed side of an angle vertex point - a fixed end point shared by two rays forming an angle x y standard position vertex

29. Ex. 1 Convert 15.735o N latitude to degrees, minutes and seconds Ex. 2 Write 32o 16' 22" as a decimal value, round to the nearest thousandth. 1. multiply the seconds by (1/3600) 2. multiply the minutes by (1/60) 3. add together.

30. In Calculator degrees minutes seconds

31. *if an angle is rotated clockwise the degree measure is negative if rotated counter-clockwise, the degree measure is positive Ex. 3 What is the angle measure given by each number of rotations? a.5.5 rotations clockwiseb.3.7 rotations counter clockwise

32. draw in standard position

33. two angles in standard position with the same terminal side are called co- terminal angles so, 55o, 415o, and -305o are all co-terminal angles

34. two angles in standard position with the same terminal side are called co-terminal angles To find all co-terminal angles for some given angle we can use the simple equation where is the degree measure of an angle Ex. 4 Find the co-terminal between 0o and 360o for each angle below a. 835ob. -1350o, where k is an integer

35. A reference angle is the acute angle formed by the terminal side of a given angle and the x- axis. 155 o What is the reference angle for 155 ? o

36. A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. Ex. 5 Find the reference angle for each degree measure below a. 120ob. 315oc. 210o

37. More trig functions! reciprocal functions aka co-functions cosecant θ = 1 sin θ secant θ = 1 cosine θ cotangent θ = 1 tangent θ co-functions

38. Ex. 2 If, find Ex. 3 If, find Ex. 3 If, find cot θ

39. Ex. 4 Find the values of the six trig functions for E: E D F 3cm 7cm

40. Ex. 4 Find the values of the six trig functions for E: E D F 3cm 7cm

41. page 280 (5-16) page 288 (5-8)

43. Example 5. The chair lift at a ski resort rises at an angle of 20.75 degrees and attains a vertical height of 1200 feet. How far does the chair lift travel up the side of the mountain? Example 6 A tree casts a 60 foot shadow. The angle of elevation of the sun is 30˚. What is the height of the tree? Example 7 Looking down from the roof of a house at an angle of 23˚ (with respect to horizontal) a shiny object is seen. The roof of the house is 32 feet above the ground. How far is the shiny object from the house?

44. 5.4 Solving Right Triangles Applications of Trigonometry use trigonometry to solve each problem rounds answers to the nearest tenth

46. d h

48. 4.5

49. (in feet) store this value (5808 tan 5.463) as "h" in your calculator The Washington Monument is about 555.5 ft tall. 5808 ft h The Lincoln Memorial and the Washington Monument are about 4224.3 ft apart.

50. Example 5. The chair lift at a ski resort rises at an angle of 20.75 degrees and attains a vertical height of 1200 feet. How far does the chair lift travel up the side of the mountain? Draw and label a diagram, and then solve the problem.

51. Angle of Elevation The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

52. Angle of Depression The object is below the level of the observer, then the angle between the horizontal and the observer's line of sight

53. An observer in the top of a lighthouse determines that the angles of depression to two sailboats directly in line with the lighthouse are 3.5 and 5.75 degrees. If the observer is 125 feet above sea level, find the distance between the boats.

54. Example 6 A tree casts a 60 foot shadow. The angle of elevation of the sun is 30˚. What is the height of the tree? shadow = 60 ft sunbeam height where does the 30 degree angle go?

55. Example 7 Looking down from the roof of a house at an angle of 23˚ (with respect to horizontal) a shiny object is seen. The roof of the house is 32 feet above the ground. How far is the shiny object from the house? where does the 23 degree angle go? 32 ft x=distance from the house

56. Assignment 5.4 p. 302 (10-18, 20, 23, 25, 26, 27) Memorize the unit circle (degrees and ordered pairs) Read section 5.5

60. coordinates of the endpoints of the edge of the circle and terminal side of the angle

61. 30-60-90 triangles1 30 60

62. 30-60-90 triangles 1 30 60

63. x x 45-45-90 triangles 1 45

65. Copy this table onto a sheet of notebook paper θ θ θ θ θ θ

66. undefined θ θθθ θ θ

67. self-assess (10 points) and turn in your homework from the book at my desk page 280 (evens 18-58) page 288 (evens 10-22) Please keep your trig table out on your desk so we can check it together

68. θ θ θθ θ θ

69. homework finish filling in the trig table do page 280 (evens 18-58) do page 288 (evens 10-22) memorize the unit circle degrees and ordered pairs

70. 5.5 Inverse Trig Functions Objective: To find the measures of the acute angles in a triangle, when two of the sides are known. make sure your calculator is in degee mode!

71. Using your knowledge of the unit circle, complete each of the following. d o these from memory without looking in your notes 1. cos(30)=_____ 2. sin(300)=_____ 3. tan(120)=_____ 4. cos(____) = 1/2

72. 1. cos(30)=_____ 2. sin(300)=_____ 3. tan(120)=_____ 4. cos(____) = 1/2 Using your knowledge of the unit circle, complete each of the following. d o these from memory without looking in your notes

73. (this one you know from your unit circle) what angles have a cosine value of Solve for θ.

74. to find the angle θ that has a sine of 2/3, you use the inverse sine function (aka arcsine) θ

75. there is also a point in quadrant 2 whose y- coordinate is 2/3!!! Use reference angles to find its measure.

76. use arccosine to solve for the angle find the reference angle for the above diagram find the other angle measure 1. 2. 3.

80. Example 5 Solve the triangle. find all the other sides and angle measures

81. Example 6 Solve the triangle. b=18, c=52

82. Assignment for 5.5 pages 309-310 * no calculators on (5, 6, 15-21); answers come from knowing the unit circle and your trig table * use calculators on (9-14, 28-30, 36-39)

85. 5.6 Use the Law of Sines to solve an oblique triangle means a not at a right angle or classified as any special angle.

86. Use the Law of Sines to solve each triangle. Round to tenths. 1. a= A= 46 b= B= c= 56 C= 63 5.6 The Law of Sines

87. 2. a= A= 65 b= B= c= 14 C= 82

88. 3. a= A= 50 b= 76 B= c= C= 33.5

89. 4. a= A= 40 b= 12 B= c= C= 22.5

90. Use the sine function to find the area of each triangle. Area = ½ the product of 2 sides times the sine of the angle between them a b C or

91. 5. Find the area of the triangle. Round to tenths.

92. Assignment for 5.6 page 316 (5-12,18-20,25-34)

94. Use the Law of Sines to solve each triangle. Round to tenths. (some have 1 solutions, some have 2 solutions, and some have no solution)

96. Round to tenths

97. hint: this is an isosceles triangle hint: set up your ratio and then cross multiply

99. Given two adjacent side lengths and the angle opposite one of them, there is no definite completion of a triangle. According to the Triangle Congruence Postulates, SSA(side side angle) is not a way triangles can be proven congruent. Why?

100. With SSA, there is a fixed angle connecting the base of the triangle and one of the adjacent side. The length of the base is unknown, the lenth of the third side is fixed, but the angles adjacent are unknown, leaving us with 2 options for angle measurements.

101. We can have 1 solution, 2 solutions, or no solution. This depends on the length of the swinging side compared to the height of the triangle. Use trig, the fixed angle and fixed height to determine.

107. 5.8 Use the Law of Cosines to solve oblique triangles

108. An oblique triangle does not have a right angle in it. It could be acute (all three angles less than 90) or it could be obtuse (one angle greater than 90 and the other two less than 90) 5.8 Use the LAW OF COSINES to solve an oblique triangle page 327 Hero's formula for area of an oblique triangle (page 330) a is the side accross from the given angle ALWAYS *semiperimeter is half of the perimeter

109. a=A= b=B= c=C= store this value for b in your calculator

110. a=A= b=B= c=C= a=A= b=B= c=C=

111. 44 47 53 find the perimeter of the triangle find the semiperimeter of the triangle example 3 Example 4 Use Hero's formula (page 330) to find the area of a triangle whose sides are 44 cm, 47 cm, and 53 cm. Round to the nearest tenth. Hero (or Heron) of Alexandria was a Greek mathematician and physicist who lived in the 1st century A.D.

112. Assignment for 5.8 pages 331 ( 10 - 28) Tues: Law of Sines 5.6 Wed: Review Thurs: 6.1 and 6.2 Friday: Test (5.5, 5.6, 5.8)