1. Welcome
to Week 1
College
Trigonometry

2. Trigonometry?
What is trigonometry?

3. Trigonometry?
Tri means three Gon means sides Ometry means measurement

4. Trigonometry?
Originated with the Egyptians

5. Trigonometry?
Arabic Trigonometry was developed in order to observe holy days on the correct days in all parts of the Islamic world

6. Trigonometry?
There was also a need for non-navigators to be able to travel to Mecca each year and return successfully

7. Trigonometry?
In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī wrote the first modern trig book (blame him…)

8. Trigonometry?
The trig functions and natural exponentials and logarithms we study in this class are needed in most formulas used to describe how our complicated universe works

9. Trigonometry?
astronomy banking electronics biology atoms forensics construction ...

10. Trigonometry!
At last! Something USEFUL!!!

11. Questions?

12. Angles
To study triangles, let’s start with one corner: an angle

13. Angles
A ray is half of a line it has an origin the other end stretches on forever
origin

14. Sun rays

15. Death rays

16. Angles
Real-life death rays: Gamma rays Cosmic rays

17. Angles
If two rays start at the same origin, they form an “angle”

18. Angles
Their point of common origin is called the “vertex”
vertex

19. Angles
Angles are usually represented by lowercase Greek letters: α, β, θ θ

20. Angles
Angles have an initial (beginning) side and a terminal (ending) side
terminal θ
initial

21. Angles
The “standard position” for an angle: vertex at origin and initial side along x-axis θ x
(0,0)

22. Angles
Positive angles - counterclockwise rotation from initial side θ

23. Angles
Negative angles - clockwise rotation from initial side θ

24. Which of the following graphs represent negative angles?
IN-CLASS PROBLEMS
Which of the following graphs represent negative angles?

25. Angles
Remember graph paper?

26. Angles
Graph paper is made of angles:

27. Angles
Graph paper is split into 4 quadrants:

28. Angles
The quadrants move counterclockwise around the grid just like positive angles II I
III IV

29. Angles
An angle with a terminal side in a quadrant "lies" in that quadrant II I
III IV

30. Which quadrant does each angle lie?
ANGLES
IN-CLASS PROBLEMS
Which quadrant does each angle lie?

31. Angles
An angle with a terminal side along any axis is called “quadrantal”

32. Angles
Angles are measure by determining the amount of rotation from the initial side to the terminal side θ

33. Angles
A degree, symbolized by ° measures angles
23° θ

34. ANGLES
IN-CLASS PROBLEMS
Which of the graphs represents the angle θ = –45º in standard position?

35. Angles
Types of angles:

36. ANGLES
IN-CLASS PROBLEMS
Classify the angle:

37. Classify the angle: 168º 42º 90º 180º 263º
ANGLES
IN-CLASS PROBLEMS
Classify the angle: 168º 42º 90º 180º 263º

38. Angles
A “reference angle” is a positive acute angle θ° formed by the terminal side of a nonacute angle θ and the x-axis

39. Which is the reference angle?
ANGLES
IN-CLASS PROBLEMS
Which is the reference angle?

40. ANGLES
IN-CLASS PROBLEMS
Co-terminal angle θ ± k•360° Which angle is co-terminal to 275º? a) 85º b) –85º c) –55º d) 55º

41. ANGLES
IN-CLASS PROBLEMS
Complementary - two positive angles whose sum = 90° Find the complementary angle to 32º a) 148º b) –32º c) 328º d) 58º

42. ANGLES
IN-CLASS PROBLEMS
Supplementary - two positive angles whose sum = 180° Find the supplementary angle to 32º: a) 148º b) –32º c) 328º d) 58º

43. ANGLES
IN-CLASS PROBLEMS
Explementary - two positive angles whose sum = 360° Find the explementary angle to 32º: a) 148º b) –32º c) 328º d) 58º

44. Questions?

45. A polygon with three sides and three angles is called _______
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
A polygon with three sides and three angles is called _______

46. Right Angle Triangles
A triangle!

47. RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
A special triangle has one corner that is a right angle – what is it called?

48. RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
The side of the triangle opposite of the right angle is called: __________

49. Right Angle Triangles

50. RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
Given the lengths of two sides of a right triangle, you can calculate the third using ____ a2 + b2 = c2

51. Right Angle Triangles
Pythagorean Theorem video

52. Find the missing side: a = 4 b = 3 c = _____ a = _____ b = 3 c = 5
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
Find the missing side:
a = b = c = _____
a = _____ b = c = 5
a = b = _____ c = 5
a = b = c = _____
a = b = c = _____
a = b = c = _____

53. Questions?

54. Trigonometry
Based on a right triangle:

55. Trigonometry
Trigonometry came to us from the ancient Greeks who drew triangles in circles

56. Trigonometry
There are six trig functions: sine = length of side opposite θ length of hypotenuse

57. Trigonometry
cosine = length of side adjacent θ length of hypotenuse

58. Trigonometry
tangent = length of side opposite θ length of side adjacent θ

59. Trigonometry
cotangent = length of side adjacent θ length of side opposite θ secant = length of hypotenuse length of side adjacent θ cosecant = length of hypotenuse

60. Trigonometry
US vs Europe
bakerfamilytree.blogspot.com

61. Trigonometry

62. Trigonometry

63. Trigonometry
Based on the formula a2+b2=c2, where c is the hypotenuse and a and b are the other sides, there are three identities: sin2 θ + cos2 θ = tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ

64. Trigonometry
Hipparchus calculated the first trig table

65. Trigonometry
Cot/Sec/Csc Stickers

66. TRIGONOMETRY
IN-CLASS PROBLEMS
Find these trig functions using your calculator: sin 10º sin 0º sin 45º sin 90º cos 20º cos 0º cos 45º cos 90º tan 30º tan 0º tan 45º tan 90º

67. Evaluate tan 30º a) 𝟑 b) 𝟐 c) –1 d) 1
TRIGONOMETRY
IN-CLASS PROBLEMS
Evaluate tan 30º a) 𝟑 b) 𝟐 c) –1 d) 1

68. Evaluate sec 45º a) 𝟑 b) 𝟐 c) –1 d) 1
TRIGONOMETRY
IN-CLASS PROBLEMS
Evaluate sec 45º a) 𝟑 b) 𝟐 c) –1 d) 1

69. Given sin θ = 3/5 and cos θ = 4/5, find tan θ a) 4/5 b) 3/4
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 3/5 and
cos θ = 4/5, find tan θ
a) 4/5 b) 3/4
c) 5/3 d) 5/4

70. Given sin θ = 2/3 and cos θ = 5 /3, find cot θ a) 1 b) 5 /2
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 2/3 and
cos θ = 5 /3, find cot θ
a) 1 b) 5 /2
c) 2/ d) 3/2

71. Given sin θ = 3/5 and cos θ = 4/5, find csc θ a) 4/5 b) 3/4
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 3/5 and
cos θ = 4/5, find csc θ
a) 4/5 b) 3/4
c) 5/3 d) 5/4

72. Given sin θ = 2/3 and cos θ = 5/3, find sec θ a) 1 b) 3/5
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 2/3 and
cos θ = 5/3, find sec θ
a) 1 b) 3/5
c) 5/3 d) 5/2

73. TRIGONOMETRY
IN-CLASS PROBLEMS
Which of the following expressions represents the same value as csc 35º?
a) sec 55º b) cos 55º
c) tan 55º d) cot 55º

74. Simplify sin2 10º + cos2 10º a) 2 b) 1 c) 3 d) 10 TRIGONOMETRY
IN-CLASS PROBLEMS
Simplify sin2 10º + cos2 10º
a) b) 1
c) d) 10

75. Trigonometry

76. TRIGONOMETRY
HOMEWORK PROBLEM
The wife of the victim said that she had just asked him for a divorce when he suddenly pulled a .357 out of his jacket pocket and shot himself in the head.

77. TRIGONOMETRY
HOMEWORK PROBLEM
According to her statement, he was standing beside the kitchen sink at the time of the shot. The victim has a single, near-contact entrance wound above his right ear 67” above the heel. The shot did not exit.

78. TRIGONOMETRY
HOMEWORK PROBLEM
Determine the height of impact to determine if the wife is telling the truth or not.

79. TRIGONOMETRY
HOMEWORK PROBLEM
Blood spatter pattern: A B C and D are the angles of impact, the other numbers are the distance to the point of convergence

80. Calculate the height of impact using the following equation:
TRIGONOMETRY
HOMEWORK PROBLEM
Calculate the height of impact using the following equation:
tangent of the distance from height of
angle of impact * base of spatter to = impact
θ pt. of convergence

81. Distance (D) to Point of Convergence
TRIGONOMETRY
HOMEWORK PROBLEM
STAIN
Angle
of Impact θ
tan(θ)
Distance (D) to Point of Convergence
(D) * tan(θ) A
62º
20.5" B
42º
34.5" C
29º
51.375" D
22º
76.25"

82. TRIGONOMETRY
HOMEWORK PROBLEM
1) Sum of (D)*tan(θ) = _____ 2) Number of Stains = _____ 3) Height of impact = 1)/2) = _____

83. TRIGONOMETRY
HOMEWORK PROBLEM
If the victim's ear was 67” above the floor, was the wife telling the truth?

84. Questions?

85. Liberation!
Be sure to turn in your exercises to me before you leave Don’t forget your lab homework due next week! Have a great rest of the week!