Chapter sixfinishes off our exploration of trigonometry with some final thoughts on some very useful applications. Some of this will be review but the final section is quite challenging. We begin by demonstrating how to solve any triangles by using the Law of Sines and Law of Cosines. We will also learn a couple of unique methods to find the area of triangles without knowing the altitude. Then we will examine vectors and some of the most common vector operations. And we will finish by covering complex numbers in trig form and how we can use DeMoivre’s Theorem to find powers and roots of complex numbers. Please ask questions regularly in class or stop by to see me or go to the MathResourceCenter in room C117 for extra help.

1.6.1Law of Sines

Pg. 398 # 3-19 odd

2.6.1Pg 398 # 21-35 odd

3.6.2Law of Cosines

Pg. 405-407# 1-13 odd, 21-25 odd, 26, 27-35 odd

4.6.2Worksheet on determining how many triangles are possible with the given information

5.6.3**Vectors in the Plane**

Pg. 417-418# 1-11 odd, 13-18 all, 19-23 odd

6.6.3Pg. 418-419# 29-65 odd

7.6.3Pg. 418-420# 69-81 odd

8.6.3Vector Application Additional Practice Worksheet & Targets 9ab, 10a-d

9.6.4**Vectors and the Dot Product**

Pg. 429-430# 1-37 odd

10.6.4Pg. 430# 39- 42 all, 47, 51, 56, 57

11.6.5**Trig Form of a Complex Number**

Pg. 440#7- 43 odd

12.6.5Pg. 441# 55, 57, 63-71 odd, 75, 79, 81, 91, 93, 95, 97, 105, 111, 113

13Pg. 443Read through Chapter Summary for sections 6.1-6.5. What did you learn?

Pg. 444-447#7, 11, 17, 18, 27, 31, 33, 39, 49, 55, 57, 61, 65, 67, 71, 87, 91, 93, 98, 107, 113, 123

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Ch 6 Test

Part I / Ch 6 Test

Part II

**Even Answers to Chapter 6**

**Section 6-2 Pg: 405**

26. Bearing a B: N43.03°E

Bearing at C: S66.95°E

**Section 6-3 Pg: 417**

14. Graph

16. Graph

18. Graph

**Section 6-4 Pg: 429**

40. w1 = <0,0>

w2 = <4,2>

u = <4,2> + <0,0>

42. w1 = 2<-1,1>

w2 = 3<-1,-1>

u = <-3,-3> + <-2,2>

56. 937.7 pounds

5318 pounds

**Chapter Review Pg: 444**

18. a = 634.7 ft

w= 586.4 ft

98. 104 pounds

H-Pre CalculusName______

6.3 Vector Applications Additional Practice

1. Three forces with magnitudes of 65 pounds, 34 pounds, and 50 pounds act on an object at angles of -20˚, 60˚, and 135˚, respectively, with the positive x axis. Find the direction and magnitude of the resultant of these vectors.

2. A ball is thrown with an initial velocity of 65 feet per second at an angle of 43˚ with the horizontal (x-axis). Find the vertical and horizontal components of the velocity.

3. Two cranes are lifting an object that weighs 19,080 pounds. Find the tension in the cable of each crane.

4. A commercial jet is flying from Miami to Seattle. The jet’s velocity with respect to the air is 580 miles per hour and its bearing is N28˚W. The wind is blowing from the southwest with a velocity of 60 mph.

a. Draw a picture.

b. Write the velocity of the wind as a vector in component form.

c. Write the velocity of the jet as a vector in component form.

d. What is the speed of the jet with respect to the ground? What is the direction of the jet?

Name ______

H-Pre-Calculus

**Chapter 6 Word Problems**

1.An airplane is traveling at a speed of 500 mph with a bearing of N40oW at a fixed altitude and no wind. As the plane crosses the Mississippi river, it encounters a wind blowing with a velocity of 50 mph in the direction of N20oE. What is the resultant speed and direction of the plane?

2.Sally is playing tug of war with two friends. She is pulling with a force of 50N at 250º Allison exerts a force of 40N at 65º. Maria exerts a force as well. To achieve equilibrium and thus, not lose the game, what is the force Maria must exert and at what angle?

3.A 10,000 pound object is suspended on a wire tied to two poles. The angle between the horizontal and the wire to the shorter pole is 22o. The angle between the horizontal and the wire to the taller pole is 49o. Find the tension in the cable to each pole.

4.A 300 pound cart sits on a ramp inclined at 25o. Assume the only fore to overcome is the force of

gravity. What is the force required to keep the cart from rolling down the ramp?

5.A truck with a gross weight of 40,000 pounds is parked on a slope of 12o. Assume the only fore to overcome is the force of gravity. Find the force required to keep the truck from rolling down the hill.

6.A mover exerts a horizontal force of 35 pounds on a crate as it is pushed up the ramp that is 20 feet long and inclined at an angle of 23o above the horizontal. Find the work done on the crate.

7.Two lighthouses are 30 miles apart along a straight shore. The ship is to the NE of one lighthouse and NW of the other lighthouse. The ship is 20 miles from one lighthouse and 15 miles from the other. How far is the ship from the shore?

8.The lengths of two adjacent sides of a parallelogram are 4.5 yards and 6.8 yards. Find the area of the parallelogram is the angle between the two sides is 25˚.

9.Sara is in a boat traveling due west parallel to the shore. At one point Sara sees her friend Ashley on the shore at a bearing of S35˚W. Sara continues west for 400 more yards, where now she sees her friend at a bearing of S27˚E. How far is Sara from Ashley at both points? How far is Sara from the shore?

10.Two forces act on an object with magnitudes of 37 pounds and 42 pounds at angles of -40˚ and 91˚, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these forces.

H-Pre-Calculus Targets

Chapter 6Additional Topics in Trigonometry

Major Objective: Apply trigonometry to solve triangles, represent vectors and perform operations with complex numbers.

Detailed Unit Objectives:

1.Use the Law of Sines and the Law of Cosines to solve oblique triangles.

2. Find areas of oblique triangles.

3.Represent vectors as directed line segments and perform mathematical operations on vectors.

4. Find direction angles of vectors.

5.Find the dot product of two vectors and use properties of the dot product .

6. Multiply and divide complex numbers written in trigonometric form.

7.Find powers and nth roots of complex numbers.

Section 6.1Law of Sines

1.I can use the Law of Sines to solve oblique triangles (AAS, ASA, or SSA).

Solve the following triangles.

a.A = 29o, B = 62o, c =11.5b.A = 63o, a = 17, b = 18

c.C = 17o, a = 15, c = 11d.A = 42o, B = 55o, a = 15

2.I can find the areas of oblique triangles.

a.a = 7.5, b = 9, C = 100ob.A = 60o, a = 12, B = 75o

3.I can use the Law of Sines to model and solve real-life problems.

a.The bearing from a fire tower to Station A is N53oE and are 45 kilometers apart. A fire is spotted from the fire tower has a bearing of N75oE from the tower and N 120oE from Station A. Find the distance of the fire from the tower and the distance of the fire from

StationA.

b.A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time the bearing to the lighthouse is S70oE, and 15 minutes later the bearing is S63oE. The lighthouse is located at the shoreline. Find the distance from the boat to the shoreline.

c.A 10-meter telephone pole casts a 17-meter shadow directly down a slope when the angle of elevation of the sun is 42o. Find the angle of elevation of the ground.

Section 6.2Law of Cosines

4.I can use the Law of Cosines to solve oblique triangles (SSS or SAS).

Solve the following triangles.

a.A = 52o, b = 6, c = 8b.a = 21, b = 16.7, c = 10.3

5.I can use the Law of Cosines to model and solve real-life problems.

a.A parallelogram has sides of 55 cm and 71 cm. Find the length of each diagonal of the parallelogram if the largest angle measures 106o.

b.Two ships leave port at 9 A.M. One travels at a bearing of N53oW at 12 miles per hour and the other travels at a bearing of S67oW at 16 miles per hour. Approximate how far apart the ships are at noon.

c.A ship travels 60 miles due east, then adjusts its course “north-eastward.” After traveling 80 miles in the new direction, the ship is 139 miles from its point of departure. What is the bearing of the “north-eastward” direction.

d.A corner of Adam’s Park occupies a triangular area that faces two streets that meet at an angle measuring 85o. The sides of the area facing the streets are each 60 feet in length. The park’s landscaper wants to plant shrubs around the edges of the triangular area. Find the perimeter of the triangular area

6.I can use Heron’s (Hero’s) Area Formula to find areas of triangles.

a.a = 14, b = 17, c = 7b.a = 4.45, b = 18.5, c = 3.1

c.A parking lot has the shape of a parallelogram. The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70o. What is the area of the parking lot?

Section 6.3Vectors in the Plane

7.Write the component form of vectors.

a.Write the component form of the vector with initial point: (5, 7) and terminal point: (12, -1)

8.Perform basic vector operations and represent vectors graphically.

Given: u = <-3, 5>, v = <4, 10> find each of the following:

a.2u + 3vb.3u - v

9.Write vectors as linear combinations of unit vectors.

a.Find a unit vector in the direction of the given vector: <3, 7>

b.Find a unit vector in the direction of the given vector: 2i – 5j

10.I can use vectors to model and solve real-life problems.

a..An airplane is traveling at a speed of 800 mph with a bearing of S40oE at a fixed altitude. Because of the wind, its groundspeed is 760 mph with a bearing of S43oE. Find the direction and speed of the wind.

b.Sally is playing tug of war with two friends. She is pulling with a force of 75N at 150º Allison exerts a force of 55N at 55º. Maria exerts a force as well. To achieve equilibrium and thus, not lose the game, what is the force Maria must exert and at what angle?

c.A 500 pound object is suspended on a wire tied to two poles. The angle between the horizontal and the wire to the shorter pole is 29o. The angle between the horizontal and the wire to the taller pole is 38o. Find the tension in the cable to each pole.

d.Use the figure below to determine the tension in each cable supporting the load.

Section 6.4Vectors and Dot Products

11.I can find the dot product of two vectors and use properties of the dot product.

Given u = and v= , find each of the following:

a.u ·vb.u · u

c.3u · 7vd.

12.I can find angles between vectors and determine whether two vectors are orthogonal.

a.u = v = b.u = v =

c.u = 5i – 7j v = 7i – 5j

13.I can find the projection of a vector onto another vector.

a.Find the projection of v onto u if v = and u .

b.Find the projection of u onto v if v = and u .

14.I can use vectors to find forces and the work done by a force.

a.A 700 pound cart sits on a ramp inclined at 28o. Assume the only force to overcome is the force of gravity. What is the force required to keep the cart from rolling down the ramp?

b.A mover exerts a horizontal force of 50 pounds on a crate as it is pushed up the ramp that is 30 feet long and inclined at an angle of 19o above the horizontal. Find the work done on the crate.

c.A mover exerts a force of 150 pounds (in a direction parallel to the ramp surface — 17o above the horizontal) on a crate in as it is pushed up the ramp that is 20 feet long and inclined at an angle of 17o above the horizontal. Find the work done on the crate.

Section 6.5Trigonometric Form of a Complex Number

15.I can find absolute values of complex numbers.

a.Find the absolute value of -3 – 7i

b.Find the absolute value of 4 – 9i

16.I can convert between standard and trigonometric forms of complex numbers.

a.Write 2 + 5i in trigonometric form.

b.Write in trigonometric form.

c.Write in standard form.

d.Write in standard form.

17.I can multiply and divide complex numbers written in trigonometric form.

a.b.

c.d.

e.(3 + 3i)(1 – i)f.

18.I can use DeMoivre’s Theorem to find powers of complex numbers.

a.(2 + 3i)5b.

c.d.

19.I can find the nth roots of complex numbers.

*Find the indicated roots of the complex number and write each of the roots in standard form.*

a.b.

c.d.

e.Find all solutions for x4 + 3i = 0

Chapter 6

Target – Answers

1a.C = 89o

a = 5.576

b = 10.155

1b.B = 70.634oB = 109.366o

C = 46.366oC = 7.634o

c = 13.809c = 2.535

1c.A = 23.496oA = 156.504o

B = 139.504oB = 6.496o

b = 24.432b = 4.256

1d.C = 83o

b = 18.363

c = 22.250

2a.33.237 units2

2b.56.784 units2

3a.23.840 km & 58.581 km

3b.3.185 miles

3c.16.078o

4a.a = 6.395

B = 47.675o

C = 80.325o

4b.A = 99.372o

B = 51.686o

C = 28.942o

5a.101.088 cm & 76.898 cm

5b.43.267 miles

5c.N76.153oE

5d.201.071 ft.

6a.47.749 units2

6b.not a triangle

6c. 6577.848

7a.

8a.

8b.

9a.

9b.

10a.N4.103oE, 57.153 mph

10b.89.056N at 292.031o

10c.Fleft = 428.032 lbs

Fright = 475.076 lbs

10d.Fleft = 1420.888 lbs

Fright = 3079.856 lbs

11a.-23

11b.29

11c.-483

11d.5

12a.58.736o

12b.90o

12c.18.925o

13a.

13b.

14a.328.630 lbs.

14b.1418.278 ft-lbs

14c.3000 ft-lbs

15a.

15b.

16a.

16b.

16c.

16d.

17a.

17b.

17c.

17d.

17e.6

17f.

18a.

18b.

18c.

18d.

19a.

19b.

19c.

19d.

19e.