1. Laws of Sines and Cosines
Sections 8.1 and 8.2

2. Objectives
Apply the law of sines to determine the lengths of side and measures of angle of a triangle.
Solve word problems requiring the law of sines.
Apply the law of cosines to determine the lengths of side and measures of angle of a triangle.
Solve word problems requiring the law of cosines.
Solve a word problem requiring Heron's formula.

3. Law of sines Law of cosines Heron’s formula
The formulas listed below will allow us to more easily deal with triangles that are not right triangles.
Law of sines
Law of cosines
Heron’s formula

4. Formulas Law of sines Law of cosines Heron’s formula or or
a, b, and c are the lengths of the sides of the triangle
P is the perimeter of the triangle
A is the area of a triangle

5. Law of Sines

6. Law of Cosines

7. Use the Law of Sines to find the value of the side x.
We are told to use the law of sines to find x. In order to use the law of sines, we need to have the lengths of two sides and the measures of the angle opposite those sides. In this case we have one side and the side we are looking for. We have the measure of the angle opposite the side we are looking for, but are missing the measure of the angle opposite the side we have.
continued on next slide

8. Use the Law of Sines to find the value of the side x.
Since we have the measures of two of the three angles, we can use the fact that the sum of the measures of the angles of a triangle add up to 180 degrees. This will give us:
Now that we have the measure of the angle opposite the side AB, we can apply the law of sines to find the value of x.
continued on next slide

9. Use the Law of Sines to find the value of the side x.

10. Use the Law of Cosines to find the value of the side x.
In order to use the law of cosines, we need the lengths of two sides and the measure of the angle between them. We have that here. We can let side a be x and angle α be the 39 degree angle. Sides b and c are the lengths 21 and 42.
continued on next slide

11. Laws of Sines and Cosines
Sections 8.1 and 8.2

12. Objectives
Apply the law of sines to determine the lengths of side and measures of angle of a triangle.
Solve word problems requiring the law of sines.
Apply the law of cosines to determine the lengths of side and measures of angle of a triangle.
Solve word problems requiring the law of cosines.
Solve a word problem requiring Heron's formula.

13. Formulas Law of sines Law of cosines Heron’s formula or or
a, b, and c are the lengths of the sides of the triangle
P is the perimeter of the triangle
A is the area of a triangle

14. Use the Law of Cosines to find the value of the side x.
Now we plug into the law of cosine formula to find x.
Since length is positive, x is approximately

15. Two ships leave a harbor at the same time, traveling on courses that have an angle of 140 degrees between them.  If the first ship travels at 26 miles per hour and the second ship travels at 34 miles per hour, how far apart are the two ships after 3 hours?
For this problem, the first thing that we should do is draw a picture. Once we have the picture, we may be able to see which formula we can use to solve the problem.
continued on next slide

16. Two ships leave a harbor at the same time, traveling on courses that have an angle of 140 degrees between them.  If the first ship travels at 26 miles per hour and the second ship travels at 34 miles per hour, how far apart are the two ships after 3 hours?
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
harbor
harbor
140°
34mph*3hr = 102 miles x
ship 1
ship 2
Looking at the labeled picture above, we can see that the have the lengths of two sides and the measure of the angle between them. We are looking for the length of the third side of the triangle. In order to find this, we will need the law of cosines. x will be side a. Sides b and c will be 78 and Angle α will be 140°.
continued on next slide

17. Two ships leave a harbor at the same time, traveling on courses that have an angle of 140 degrees between them.  If the first ship travels at 26 miles per hour and the second ship travels at 34 miles per hour, how far apart are the two ships after 3 hours?
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
26mph*3hr = 78 miles
harbor
ship 1
ship 2
140°
34mph*3hr = 102 miles x
harbor
harbor
140°
34mph*3hr = 102 miles x
ship 1
ship 2
Since distance is positive, the ships are approximately miles apart after 3 hours.

18. Approximating the area of a triangle
Heron’s Formula
where P is the perimeter of the triangle and a,b, and c are the lengths
of the sides of the triangle. OR
The area of a triangle equals one-half the product of the lengths of any two sides and the sine of the angle between them.
i.e. 1/2ab sinγ = A

19. 48°20′
37.0 cm
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7-19

20. 8.3 Vectors, Operations, and the Dot Product
Basic Terminology ▪ Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ Dot Product and the Angle Between Vectors
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7-20

21. 8.3 Vectors
A vector is an object that has a magnitude and a direction.
Given two points P1: and P2: on the plane, a vector v that connects the points from P1 to P2 is v =
i j.
Unit vectors are vectors of length 1.
i is the unit vector in the x direction.
j is the unit vector in the y direction.
A unit vector in the direction of v is v/||v||
A vector v can be represented in component form
by v = ai + bj.
The magnitude of v is ||v|| =
Using the angle that the vector makes with x-axis in standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j 21

22. Magnitude and Direction Angle
Find the magnitude and direction angle for
Magnitude:
Direction angle:
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23. Horizontal and Vertical Components
Vector v has magnitude 14.5 and direction angle 220°. Find the horizontal and vertical components.
Horizontal component: –11.1
Vertical component: –9.3
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24. Example 4 Finding the Magnitude of a Resultant
Two forces of 32 and 48 newtons act on a point in the plane. If the angle between the forces is 76°, find the magnitude of the resultant vector.
because the adjacent angles of a parallelogram are supplementary.
Law of cosines
Find square root.
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25. Let u = 6, –3 and v = –14, 8. Find the following.
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26. Unit Vector has the same direction as a given vector, but is 1 unit long
Unit vector = (original vector)/length of vector
Simply involves scalar multiplication once the length of the vector is determined (recall the length = length of hypotenuse if legs have lengths = a & b)
Given vector, v = -2i + 7j, find the unit vector:

27. Review Answers

28. 8.4 Dot Product Objectives Find dot product of 2 vectors
Find angle between 2 vectors
Use dot product to determine if 2 vectors are orthogonal
Find projection of a vector onto another vector
Express a vector as the sum of 2 orthogonal vectors
Compute work.

29. 8.4 Vector Operations
Scalar multiplication: A vector can be multiplied by any scalar (or number).
Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j.
Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the
angle between the two vectors. v θ w
Addition/subtraction of vectors: Add/subtract same components.
Example Let v = 5i + 4j, w = –2i + 3j.
v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.
3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j. 29

30. Definition of Dot Product
The dot product of 2 vectors is the sum of the products of their horizontal components and their vertical components
Example: Let v = 5i + 4j, w = –2i + 3j.
v · w = (5)(–2) + (4)(3) = – = 2.

31. Find the dot product of v&w if v=3i+j and w= -2i - j7 -5 -7 -4

32. Properties of Dot Product
If u,v, & w are vectors and c is scalar, then

33. Angle between vectors, v and w

34. Find the angle between a=<4,-3> and b=<1,2>.

35. Parallel Vectors
Parallel: the angle between the vectors is either 0 (the vectors on top of each other) or 180 (vectors are in opposite directions).
a and b are parallel if θ = 0 or θ = ∏.

36. Perpendicular Vectors
Two vectors v and w are orthogonal (perpendicular) iff v · w = 0
Two vectors v and w are orthogonal (perpendicular) if the angle between them is θ=∏/2.

37. Let a = 1/2i – 3j and b = -2i+12j. Show that a and b are parallel.

38. Show that the pair of vectors is orthogonal. 2i+3j; 6i-4j

39. Formula for compba: If a and b are nonzero vectors, then compba = a • b

40. If c = 10i+4j and d = 3i-2j, find compdc and compcd and illustrate.

41. Work done by a force F moving an object from A to B
The work W done by a constant force a as its point of application moves along a vector b is W = a • b.

42. Trigonometric Form of Complex Numbers
Lesson 8.5

43. Graphical Representation of a Complex Number
Graph in coordinate plane
Called the complex plane
Horizontal axis is the real axis
Vertical axis is the imaginary axis
3 + 4i •
-2 + 3i •
• -5i 43

44. Absolute Value of a Complex Number
Defined as the length of the line segment
From the origin
To the point
Calculated by using Pythagorean Theorem
3 + 4i • 44

45. Find That Value, Absolutely
Try these
Graph the complex number
Find the absolute value 45

46. Trig Form of Complex Number
Consider the graphical representation
We note that a right triangle is formed
a + bi • r b θ a
How do we determine θ? 46

47. Trig Form of Complex Number
Now we use and substitute into z = a + bi
Result is
Abbreviation is often 47

48. Try It Out Given the complex number -5 + 6i Given z = 3 cis 315°
Write in trigonometric form
r = ?
θ = ?
Given z = 3 cis 315°
Write in standanrd form
a = ?
b = ? 48

49. Product of Complex Numbers in Trig Form
Given
It can be shown that the product is
Multiply the absolute values
Add the θ's 49

50. Quotient of Complex Numbers in Trig Form
Given
It can be shown that the quotient is 50

51. Try It Out Try the following operations using trig form
Convert answers to standard form 51

52. 8.6 Complex Numbers in Polar Form: DeMoivre’s Theorem
Objectives
Plot complex numbers in the complex plane
Fine absolute value of a complex #
Write complex # in polar form
Convert a complex # from polar to rectangular form
Find products & quotients of complex numbers in polar form
Find powers of complex # in polar form
Find roots of complex # in polar form

53. Complex number = z = a + bi
a is a real number
bi is an imaginary number
Together, the sum, a+bi is a COMPLEX #
Complex plane has a real axis (horizontal) and an imaginary axis (vertical)
2 – 5i is found in the 4th quadrant of the complex plane (horiz = 2, vert = -5)
Absolute value of 2 – 5i refers to the distance this pt. is from the origin (continued)

54. Find the absolute value
Since the horizontal component = 2 and vertical = -5, we can consider the distance to that point as the same as the length of the hypotenuse of a right triangle with those respective legs

55. Expressing complex numbers in polar form
z = a + bi

56. Express z = -5 + 3i in complex form

57. DeMoivre’s Theorem [r (cos θ + i sin θ)]n = rn (cos n•θ + i sin n•θ)
Ex: Change (1 + i)20 to the form a + bi

58. Taking a root (DeMoivre’s Theorem)
Taking the nth root can be considered as raising to the (1/n)th power
Now finding the nth root of a complex # can be expressed easily in polar form
HOWEVER, there are n nth roots for any complex number & they are spaced evenly around the circle.
Once you find the 1st root, to find the others, add 2pi/n to theta until you complete the circle

59. DeMoivre’s Theorem and nth roots:
Wk = n√r [cos (θ+2∏k ) + i sin (θ + 2∏k)
n n
where k = 0,1,2,…n-1.

60. Example:
Find the fourth roots of -8-8√3i.

61. Example: Find the 6th roots of z= -2 + 2i
If you’re working with degrees add 360/n to the angle measure to complete the circle.
Example: Find the 6th roots of z= i
Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

62. Example: Find the 6th roots of z= -2 + 2i
If you’re working with degrees add 360/n to the angle measure to complete the circle.
Example: Find the 6th roots of z= i
Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

63. Example: Find the 6th roots of z= -2 + 2i
If you’re working with degrees add 360/n to the angle measure to complete the circle.
Example: Find the 6th roots of z= i
Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

64. Formulas for index cards:
Law of Sines: Sin A = Sin B = Sin C
a b c
Law of Cosines:
a2 = b2 + c2 – 2bc Cos A
a2-b2-c2 = Cos A
-2bc
Area of a Triangle
A = ½ ab Sin C
A = √(s(s-a)(s-b)(s-c)); s = ½ (a+b+c)

65. a1 = ||a||cos θ a2 = ||a||sin θ Unit Vector • a
||a|| = √(a12 + a22)
a1 = ||a||cos θ a2 = ||a||sin θ
Unit Vector • a
cos θ = a•b angle between vectors
||a||||b||
compba = a•b
||b||

66. Products and Quotients of Complex #’s
z1z2 = r1r2(cos(θ1+ θ2)+isin(θ1+ θ2))
z1 = r1 (cos(θ1- θ2)+isin(θ1- θ2))
z2 r2
DeMoivre’s Theorem
[r(cos θ + isin θ)]n = rn(cosnθ + isin nθ)
nth roots
Wk = n√r [cos (θ +2пk) + isin (θ +2пk)]
n n
where k = 0,1,…n-1

67. Dot Product <a1,a2> • <b1,b2> = a1b1 + a2b2
Work Done = a • b (force * distance)

68. Review Test Answers