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Math Analysis H WS 1.1- Four Word Problems Do all work neatly on a separate sheet of paper.

1. The Stratosphere in Las Vegas, Nevada, features two thrill rides located near the top. At a particular time of day the shadow cast by the tower is 535.8 ft long. The angle of elevation of the sun is 65°. What is the height of the Stratosphere Tower?

2. The chair lift at a ski resort rises at an angle of 20.75° and attains a vertical height of 1200 feet. a. How far does the chair lift travel up the side of the mountain? b. How far from the center of the mountain's base is the bottom of ski lift?

3. A regular octagon has a perimeter of 56 inches. Find the apothem of the octagon.

4. When Mount Saint Helens erupted in 1980, the top of the mountain was blown off. A surveyor determined the height of the new summit by measuring the angle of elevation to the top to be 37°46'. She then moved 1000 feet closer to the volcano and measured the angle of elevation to be 40°30'. Find the new height of Mount Saint Helens.

--------------------------------------------------------------------------------------------------------- Math Analysis H WS 1.2- Trig Practice Do all work neatly on a separate sheet of paper.

1. If right triangle ABC had legs 36a and 77b with hypotenuse 85c , what is the measure of angle A ? a. 25 b. 35 c. 65 d. 42

2. A ladder leans against a building forming an angle of 54° with the ground. The base of the ladder is 7 feet from the building. Determine the length of the ladder. a. 1.5 ft. b. 9.4 ft. c. 8.7 ft. d. 11.9 ft.

3. A tree casts a shadow of 27 meters when the angle of elevation of the sun is 26°. Find the height of the tree to the nearest meter. a. 24 m b. 15 m c. 320 m d. 13 m

4. Which angle is not coterminal with 421°? a. 781 b. 299 c. 61 d. 119

5. Convert 288° to radians.

a. 45 b. 16

5 c. 16

15 d. 8

5

6. On a Ferris wheel, you travel through a central angle of 212 before stopping. If the

radius of the Ferris wheel is 80 feet, how many feet have you traveled?. a. 2638.9 ft b. 2838.9 ft c. 2538.9 ft d. 2938.9 ft

7. Find cot if (8, 15) is a point on the terminal side of .

a. 1517

b. 817

c. 815

d. 158

8. is an angle in standard position with point 4, 2P on the terminal side. Which statement is not correct?

a. 5cos5

b. cot 2 c. 5sin5

d. 5sec2

9. What are the values of sin and cos for the acute angle in standard position if 2tan5

?

a. 2 3sin , cos3 5

b. 3 3sin , cos2 5

c. 2 5sin , cos3 3

d. 5 2sin , cos3 3

10. Which single expression is equivalent to sin . a. cos b. sin c. sin d. cos

11. Evaluate the expression 7cos4

.

a. 2

2 b.

22

c. 0 d. 1

12. Evaluate the expression 257tan

4

.

a. 1 b. 2

2 c. 1 d.

22

13. Find one positive angle and one negative angle that are coterminal with an angle of

−328° in standard position. 14. Find the degree of the angle in standard position formed by rotating the terminal side

by 2245

of a circle.

15. Convert 74 to degrees.

16. A wheel is 5 feet in diameter and rotates at 1100 rpm.

a. What is the angular speed of the wheel? b. How fast is a point on the circumference of the wheel traveling in feet per

minute? In miles per hour?

17. Express cos csc in terms of tan

18. Evaluate cos sin6 3 without using a calculator.

Math Analysis H WS 1.3- Trig Laws Word Problems Do all work neatly on a separate sheet of paper. 1. One angle of a rhombus is 38° 42’ and its sides are 4.836 in long. Find the

length of the shorter diagonal.

2. Two observers A and B are in lighthouses 7 miles apart. Observer A spots a boat C and notes that m BAC is 64°. At the same time observer B notes that m ABC is 43°. What is the distance from B to C?

3. A ladder 27 ft long makes an angle of 58° with the horizontal when it reaches a certain window. What angle will a 33 ft long ladder make with the horizontal when it reaches the same window?

4. The radius of a circle is 12 in. What is the central angle that intercepts a chord 18 in long?

5. Two planes leave an airport at the same time. One flies SE at 186 mph, the other at a bearing of 60° at 213 mph. How far apart are they after 2 hrs?

-------------------------------------------------------------------------------------------------------- Math Analysis H WS 1.4- Trig Laws Review Do all work neatly on a separate sheet of paper. 1. Given a = 5, b = 8, and C = 70°, find c.

2. Given c = 4, a = 6, and A = 50°, find C.

3. Given A = 31°21’, C = 65°50’, and c = 6, find b.

4. Given c = 6, B = 61°40’, and A = 92°30’, find the area of the triangle.

5. Given b = 6, a = 3, and A = 64°, find the area of the triangle.

6. Given a = 20, b = 50, and c = 60, find the area of the triangle.

7. Determine the number of triangles that exist with the given information and state why. a. B = 40°, b = 30, c = 20 b. B = 140°, c = 30, b = 20 c. C = 55°10’, b = 480, c = 628 d. A = 28°, a = 4.8, b = 6

8. In a parallelogram with side lengths of 92 in and 48 in, the angle between the longer diagonal and the shorter side is 43°. What is the measure of the obtuse angle in the parallelogram?

9. Two chords measuring 18.64cm and 14.32cm intersect at a point on a circle at an angle of 114°26’. A third chord connects the noncommon endpoints of the chords to form a triangle. Find all the measurements of the triangle.

10. A triangular plot of land has two sides that measure 185ft and 147ft which intersect at an angle measuring 51°10’. Determine the area of the plot.

11. Three circles are tangent to each other with radii of 115, 150, and 225. Draw the triangle formed by connecting their centers. Determine the measure of each angle of the triangle.

12. A plane flying due east at 100 m/s is blown by a strong wind blowing due south at 40 m/s. Find the speed and bearing of the plane.

Math Analysis H WS 1.5- Review Do all work neatly on a separate sheet of paper. 1. Construct a triangle for which the terminal side lies in Quadrant IV and

24tan7

. Then find csc .

2. Find the exact values of the six trig functions if 116 .

3. Verify 2cos90 2cos 45 1 4. Evaluate the following without a calculator:

a. 23cos cos sin sin2 6 3 6

b. sec 0 2sin

c. 3csc 4sec2 2

5. If the dimensions of a rectangle are 674 ft by 106 ft, find the measure of the

angle between the shorter side and the diagonal. 6. If the angle of elevation of the top of a building from a point 63 m from the

base of the building is 56º 20', what is the height of the building? 7. Sketch each angle in standard position and give the measure of its reference

angle: a. 17 b. 185

c. 56

8. Given that the terminal side of an angle θ in standard position goes through

(-2, √5), find: a. cot b. csc

9. If the central angle is 50º lies on a circle of radius 5, find the length of the arc.

Math Analysis H WS 1.X- Trig Chart Complete the chart with the exact values of the six trigonometric functions for the given angle measurements.

Radian Degree sin θ cos θ tan θ csc θ sec θ cot θ

0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360°

x

y

x

y

Math Analysis H WS Trig Packet 1- Graphs of Sine and Cosine You should complete the graphs without the aid of a graphing calculator. 1. Sketch a graph of siny x . Use your graph to answer the following questions about the sine function.

a) What is the domain of the sine function?

b) What is the range of the sine function?

c) Where are the x-intercepts located?

d) Where is the y-intercept?

e) What is the maximum value of the graph? Where do the maximums occur?

f) What is the minimum value of the graph? Where do the minimums occur?

2. Sketch a graph of cosy x . Use your graph to answer the following questions about the cosine function.

a) What is the domain of the cosine function?

b) What is the range of the cosine function?

c) Where are the x-intercepts located?

d) Where is the y-intercept?

e) What is the maximum value of the graph? Where do the maximums occur?

f) What is the minimum value of the graph? Where do the minimums occur?

3. We sometimes refer to the point on the graph where 0x as the “starting point” of the graph. What is the starting

point of the cosine graph? _____________________ Of the sine graph? ________________________

4. Tell whether each statement below describes a characteristic of the sine function, the cosine function, both functions,

or neither function:

a) The function passes through 0, 0

b) The function is increasing on the interval 02

x

c) The period of the function is

d) The function is symmetric with respect to the x-axis

e) The function is symmetric with respect to the y-axis

f) The x-intercepts occur at multiples of

g) The x-intercepts occur at multiples of 2

h) The range of the function is 0 1y

i) The maximum values of the function occur when the x values are multiples of 2

j) The minimum value of the function is -1

k) The function decreases on the interval 0 x

5. Graph siny x and cosy x using a graphing calculator with a window that shows 2 2x .

a) Use your calculator to find all values x for which sin 0.8x

b) Use your calculator to find all values x for which cos 0.6x

c) For what values 2 2x is sin cosx x ?

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS 2.1- Trig Solving and Graphing Find all values of , from 0 360 , for which the equation is true.

1. 2sin2

2. sin 0 3. 1cos2

Find all values of , from 0 2 , for which the equation is true.

4. 3sin2

5. cos 0 6. 3cos2

Find ALL values of , for which the equation is true.

7. sin 1 8. cos 1 9. 1cos2

10. 2cos2

Graph each function

11. 5sin 1f 12. 2sin 13

f

13. 4cos 2 45 3f

Math Analysis H WS Trig Packet 2- Equations from Graphs

For each graph, determine the values of D, A, B, and C. Then, write an equation in the form siny A B x C D and

cosy A B x C D . Sine Equation Cosine Equation

1. A = B = CSin = CCos = D =

2. A = B = CSin = CCos = D =

3. A = B = CSin = CCos = D =

4. A = B = CSin = CCos = D =

Sine Equation Cosine Equation

5. A = B = CSin = CCos = D =

6. A = B = CSin = CCos = D =

7. A = B = CSin = CCos = D =

8. A = B = CSin = CCos = D =

Math Analysis H WS Trig Packet 3- Word Problems

Do work on a separate sheet. Use graph paper when necessary. 1. A leaf floats on the water bobbing up and down. The distance between its highest and lowest point is 4 cm. It moves

from its highest point down to its lowest point and then back to its highest point every 10 seconds. Write a cosine

function to model the movement of the leaf in relationship to its equilibrium point.

2. Use the data and graphs from the Ferris Wheel Problem (40 foot diameter, lowest seat is 5 feet off the ground, one

revolution every 2 minutes) to write an equation for the height of a seat as a function of time when

a) The seat starts at the bottom of the wheel

b) The seat starts at the top of the wheel.

c) Use your equations to find the height of the wheel (in both cases) after 384 seconds.

3. Draw a sketch of a graph and write a sine function to model the oscillation of tides in Savannah, Georgia if the

equilibrium point is 4.24 feet, the amplitude is 3.55 feet, the phase shift is -4.68 hours, and the period is 12.40 hours.

4. There has been an earthquake in Alaska. A tidal wave caused by this earthquake is heading toward Carmel, CA.

The water level at shore first goes down from its normal level then rises an equal distance above its normal level; the

water then returns to normal. Assume the depth of the water varies sinusoidally with time as the wave passes.

Suppose the wave has a period of 20 minutes and amplitude of 12 meters; the normal depth at the dock in Carmel is

15 meters.

a) Sketch a graph of the depth of the water at the dock as a function of time

b) Write an equation for your graph

c) Find the water height at the dock at 5 minutes 20 seconds and at 12 minutes and 15 seconds.

5. The mean average temperature in Buffalo, New York is 47.5 . The temperature fluctuates 23.5 above and below

the mean temperature. If t = 1 represents January, the phase shift of the sine function is 4.

a) Write a model for the average monthly temperature in Buffalo.

b) Use your model to predict the average temperature in March.

c) Use your model to predict the average temperature in August.

6. Below is a table of average monthly temperatures at Wellington Airport from 1971-2000. Your task is to create a

model of the data to predict the times during the year that a location would be pleasant to visit. This may be when the

average monthly temperature is over 14°C.

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

°C 17.8 17.9 16.6 14.4 12.0 10.2 9.5 9.9 11.3 12.9 14.5 16.4

a) Graph this data.

b) Write a trigonometric equation using the cosine function that best models this situation.

c) Rewrite the equation using the sine function.

d) In which hemisphere do you think Wellington Airport is located?

7. Below is a table of average monthly temperatures in Thousand Oaks, CA

Month Jan Feb March Apr May June July Aug Sept Oct Nov Dec

Temp 69 70 73 78 83 88 95 97 93 84 75 68

a) Graph this data.

b) Write a trigonometric equation using the cosine function that best models this situation.

c) Rewrite the equation using the sine function.

8. The table below contains the times that the sun rises and sets in the middle of

each month in New York City. Suppose the number 1 represents the middle of

January, the number 2 represents the middle of February, etc.

a) Find the amount of daylight hours for the middle of each month

b) Determine the amplitude, period, vertical shift, and phase shift for a

sinusoidal function that models the daylight hours.

c) Write a function to model the daylight hours

9. Geraldine rides her bicycle along a flat road at night. Ian can clearly see a reflector on the spoke of her wheel

rotating. He measures the height of the reflector above the ground at different times using a video.

Time (s) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Height (cm) 19 17 38 62 68 50 24 15 31

a) Graph this data.

b) Write a trig function to model the data

c) Find the diameter of Geraldine’s wheel

d) Determine the speed that Geraldine was traveling on her bike.

Month Sunrise AM Sunset PMJan 7:19 4:47 Feb 6:56 5:24 March 6:16 5:57 Apr 5:25 6:29 May 4:44 7:01 June 4:24 7:26 July 4:33 7:28 Aug 5:01 7:01 Sept 5:31 6:14 Oct 6:01 5:24 Nov 6:36 4:43 Dec 7:08 4:28

Math Analysis H WS Trig Packet 4- Summery of Graphs

siny x cosy x tany x cscy x secy x coty x

1 Draw one cycle of the graph

2 Period in radians

3 Domain of the function

4 Range of the function

5 x-intercepts for 0 2x

6 y-intercept

7 Maximum value

8 Minimum value

9 Intervals where increasing in 0 2x

10 Intervals where decreasing in 0 2x

11 Equations of any asymptotes in 0 2x

Math Analysis H WS Trig Packet 5- Graphing Review Complete the table below. Then graph each of the equations. 1. 4sin 3 1y x

2. 5 tan 2 3y x 3. 2sec 1y x

Period Horizontal Shift Vertical Shift Amplitude Maximum Minimum Domain Range Number of cycles from 0 to 2π

Intervals from 0 to 2π where the graph is increasing

1.

2.

3.

Complete the table below. Then graph each of the equations.

4. 3cos 2y x 5. cot3

y x

6. csc 2y x

Period Horizontal Shift Vertical Shift Amplitude Maximum Minimum Domain Range Number of cycles from 0 to 2π

Intervals from 0 to 2π where the graph is increasing

4.

5.

6.

Math Analysis H WS Trig Identities 01- Algebra Review Section A: Factor the following completely.

1. 2 6x x 2. 2sin sin 6x x 3. 2 22x xy y 4. 2 2sin 2sin cos cosx x x x 5. 4 4x y 6. 4 4sin cosx x 7. 3 3x y 8. 3 3sin cosx x 9. 2x xy 10. 2sin sin cosx x x 11. 2xy x 12. 2cos sin cosx x x

Section B: Simplify the following completely.

13. 2 2x yx y

14. 2 2tan sec

tan secx xx x

15. 1 1x y 16. 1 1

sin cosx x

17. x yy x 18. cos sin

sin cosx xx x 19. 1 x

x 20. 1 sin

sinx

x

21. 1xx

22. 1tantan

xx

23. 2

3

xx y

x xyx

24. 2

tansin cos

sin sin coscos

xx x

x x xx

Section C: Complete the square.

Section D: Simplify the left side until it equals the right side.

25. 21 2 ___ _______x 26. 1 2 2y x xy x yx y

27. 21 2sin ___ _______x 28. 1 2 2sin cos sin cos sin coscos sin

x x x x x xx x

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS Trig Identities 02- Simplifying Completely simplify each problem. The answer to each problem will be one of the following: sin x , cos x , tan x , or 1.

1. 2 21 1

sec cscx x 2. 2 2sec 1 sinx x

3. sec sin tanx x x 4. sin cos tan cot sec cscx x x x x x 5. sin secx x 6. 2 2csc 1 cosx x

7. 1 tan1 cot

xx

8. 2

2csc 1

cotxx

9. 2 21 tan 1 sinx x 10. sectan cot

xx x

11.

sin tantan csc cot

x xx x x

12. 4 2 2 4cos 2cos sin sinx x x x

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS Trig Identities 03- Proofs 1 Prove the following. Use a two-column format, showing all work.

1. 2 21 1 1

sec cscx x 2. 2 2 2cos tan cos 1x x x

3. 21 cos csc cot1 cos

x x xx

4. sec sin cot

sin cosx x xx x

5. sin 1sec tan cot

xx x x

6. csc sec cot tanx x x x

7. 2 2sin cot 1 sin 1 sinx x x x 8. cos cos 2sec1 sin 1 sin

x x xx x

9. cos 1 sin1 sin cos

x xx x

10. sec 1 tan

tan sec 1x xx x

Math Analysis H WS Trig Identities 04- Trig Review 1. A rescue team 1000 ft. away from the base of a vertical cliff measures the angle of elevation to the top of the cliff to be

70°. A climber is stranded on a ledge. The angle of elevation from the rescue team to the ledge is 55°. How far is the stranded climber from the top of the cliff?

2. Angie sees a hot air balloon in the sky from her spot on the ground with an angle of elevation of 40°. If she steps back

200 ft the new angle of elevation is 10°. If Angie is 5.5 ft tall, how far off the ground is the hot air balloon? 3. David puts a rock in his sling and starts whirling it around. He realizes that in order for the rock to reach Goliath, it

must leave the sling at a speed of 60 feet per second. So he swings the sling in a circular path of radius 4 feet. What must the angular velocity be in order for David to achieve his objective?

4. Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. If the 2-inch pulley

is rotating at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. 5. Draw the first quadrant of the unit circle and mark and label all special angles and their points. 6. Find the exact values of the following:

a. 3sin4

b. 29cos

6

c. tan(420°) d. 9sec4

e. csc(510°) f. cot 3

7. Solve the triangles:

a. Given 30, 50, 60b c C b. Given 12, 5, 20a b B 8. During a hike, hikers start at point A and head in a direction 30° west of south to point B. They hike 6 miles from point

A to point B. From point B, they hike to point C and then from point C back to point A, which is 8 miles directly north of point C. How many miles did they hike from point B to point C?

9. State the amplitude, period, phase shift, and vertical shift for the following functions:

a. 2sin 3 2 5f x x b. 63 tan 4 25

f t t

10. Write an equation to model each:

a. b. The number of minutes of daylight

Month Daylight (in h:mm) Jan 9:55 Feb 10:32 Mar 11:29 Apr 12:35 May 13:35 Jun 14:18 Jul 14:24 Aug 13:49 Sep 12:51 Oct 11:48 Nov 10:46 Dec 10:02

Math Analysis H WS Trig Identities 05- Proofs 2 Prove the following. Use a two-column format, showing all work.

1. 2 2 2 2sin tan sin tanx x x x 2. 2 2 2cos 1 tan 1 tanx x x

3. 21 cos csc sinx x x 4. sectancsc

xxx

5. 4 4 2 2cos sin cos sinx x x x 6. 2 2 2 2cot cos cos cotx x x x

7. sec sin 2 tancsc cos

x x xx x 8. 1 cos sin 2csc

sin 1 cosx x x

x x

9. sec csc csc1 tan

x x xx

10. cot 1 csc

1 tan secx x

x x

11. 1 sec tancsc sin

x xx x

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS Trig Identities 06- Review Day 1 Prove each problem is an identity. Use a two-column format, showing all work.

1. sec sin tanx x x 2. tan sin cot cos secx x x x x 3. 2 2csc 1 cos 1x x

4. 2 2cos 1 tan 1x x 5. sin cot cos2cot

sinx x x

xx

6. 4 4 2 2cos sin cos sinx x x x

7. cos 60 sin 30A A 8. cos2 1cos 1

2 cos 1xxx

9. sin cos cot cscx x x x

10. 2 2sec 1 sin 1x x 11. sin tan cos secx x x x 12. cos csc sec cot 1x x x x

13. sin cot cos2cot

sinx x x

xx

14. 2

22

1 sin cot1 cos

x xx

15. 1 tantan 45

1 tanxxx

16. cos2cos sincos sin

AA AA A

17. cos csc cotx x x 18. sin sec csc tan 1x x x x

19. 2 2 22cos sin 1 3cosx x x 20. sec cos sin tanx x x x 21. 2

22

1 tan csctan

x xx

22. 1 sin coscos 1 sin

x xx x

23. 1csc 2 sec csc

2 24. 1 sec sin1 sin2

2 secA AA

A

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS Trig Identities 07- Review Day 2 Prove each problem is an identity. Use a two-column format, showing all work.

1. tan sin sec 2. tan cot sec csc 3. 4 4 2 2csc cot csc cot

4. 2 21 sin 1 tan 1 5. cot cos csc 6. cotcsccos

7. 4 4 2 2sec tan tan sec 8. 2cos sin csc

sin

9. 2 2cot csc 1

10. 2 2

22

sin cosseccos

11. 2 21 tan sec 2 tan 12. 2

22

1 costancos

13. cos sin1 tancos

14. sec tan sec tan 1 15. tan cot sin cossec csc

16. 2

21 2sin 3sin 1 sin

cos1 3sin

x x xx

x

17.

21 sin sin cos tancosx x x x

x

18. 2tan tan cot sec

19. 2sin sin 2 tancsc 1 csc 1

x x xx x

20. 3 3

2sin cos sec sin 0

tan 11 2cos

Math Analysis H WS 4.1- Evaluating Trigonometric Functions On a separate sheet of paper, complete the following:

a) Give the degree measure(s), from 0 360x , for which the following expressions are true. b) Give the radian measure(s), from 0 2x , for which the following expressions are true.

1. 3sin2

x 2. 1cos2

x 3. cos 1x 4. cos 0x 5. tan 1x

6. tan 0x 7. 2cos

2x 8. os 1c x 9.

1sin2

x 10. 2sin

2x

11. tan 3x 12. 3tan

3x 13. sin 0x 14.

3sin2

x 15. 3tan

3x

16. tan 3x 17. 1cos2

x 18. sin 1x 19. 3cos2

x 20. sin 1x

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS 4.2- Solving Trig Equations Graphically On a separate sheet of paper, solve 5 of the following equations graphically on the interval 2 , 2 :

1. 4sin 2 3cos 2 2x x 2. 5sin 6cos 1x x

3. 33sin 2 2cos 4. 2sin 2 3cos 2 2 0

5. tan 5sin 1 6. 22cos sin 1 0

7. 3cos 3cos 1 0 8. tan 3cos

9. 4 3cos 3cos cos 1x x x 10. sec tan 3x x ------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS 4.3- Solving Trig Equations Algebraically Solve each equation for principal values of x. Express solutions in degrees. Give exact values.

Solve each equation for 0 360x . Give exact values.

1. cos 3cos 2x x 2. 22sin 1 0x 3. 2sec tan 1x x 4. cos2 3cos 1 0x x Solve each equation for 0 2x . Give exact values. 5. 24sin 4sin 1 0x x 6. cos2 sin 1x x

Solve each equation for all real values of x (in Radian). Give exact values. 7. 3cos2 5cos 1x x 8. 22sin 5sin 2 0x x 9. 23sec 4 0x 10. tan tan 1 0x x 11. An airplane takes off from the ground and reaches a height of 500 feet after flying 2 miles. Given the formula

tanH d d , where H is the height of the plane and d is the distance (along the ground) the plane has flown, find the angle of ascent at which the plane took off.

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS 4.X- Extra Practice Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval 0, 2

1. 23sin 8sin 3 0x x 2. 25cos 6cos 8x x 3. 22 tan 5 tan 3 0x x 4. 23sin 2sin 5x x 5. cot cos cosx x x 6. tan cos cosx x x 7. cos csc 2cosx x x 8. tan sec 3 tan 0x x x 9. 2sin 2sin 2 0x x 10. 24cos 2cos 1x x 11. 26sin 4sin 1x x 12. 2 2cos sin sin 0x x x

Math Analysis H WS 4.4- Inverse Trig Applications

1. The equation 254.5 23.5sin -6 3

y x

models the average monthly temperatures of Springfield, MO, where x denotes the

number of months with January represented by 1. During which two months is the average temperature 54.5°? 2. The average power P of an electrical circuit with alternation current is determined by the equation cosP VI x , where V is the

voltage, I is the current, and x is the measure of phase angle. A circuit has a voltage of 122 volts and a current of 0.62 amperes. If a circuit produces an average of 7.3 watts of power, find the measure of the phase angle.

3. Malus' Law describes the amount of light transmitted through two polarizing filters. If the axes of the two filters are at an angle of x

radians, the intensity I of the light transmitted through the filters is determined by the equation 2cosoI I x , where oI is the intensity of the light that shines on the filters. At what angle should the axes be held so the one-eighth of the transmitted light passes through the filters?

4. The strength of a magnetic field is called magnetic induction. An equation for magnetic induction is 1sinB F IL x , where F is a

force on a current, I, which is moving through a wire of length L at an angle of x to the magnetic field. A wire within a magnetic field is 1m long and carries a current of 5 amps. The force on the wire is 0.2 newton, and the magnetic induction is 0.04 newton per amperemeter. What is the angle of the wire to the magnetic field?

5. A TV camera on the ground follows the launch of a 120-foot rocket. The camera is x feet from the launch pad and the base of the

rocket is y feet above the ground. The angle, A, of the camera lens is given by the equation 2 2

120tan120

xAx y y

. Find the

angle needed for a rocket 200 feet above the ground if the camera is located 400 feet from the launch pad. 6. A nautical mile is equal to an arc length of one minute of a degree. The actual length varies slightly since the Earth is not a perfect

sphere. The formula for the length of a nautical mile in feet, L, on the latitude line x is: 6077 31cos 2L x .

a. Solve the formula for x. b. Find the length of a nautical mile on the 40th latitude. c. On what latitude is the length of a nautical mile 6060 feet?

7. A civil engineer is designing a curve for a new highway. She uses the equation 2

tan vxgr

, where x is the angle the curve should

be banked, r is the radius of the circular arc and g is gravity (32 ft/s2) and v the velocity of the vehicle. At what angle should a curve with a radius of 1200 feet be banked, to accommodate a speed of 65 mph?

------------------------------------------------------------------------------------------------------------------------------------------------------------------ Math Analysis H WS 4.5- Solving Trig Equations Review Solve graphically. Find all solutions. 1. 3 2cos 3cos 2sin 1x x x 2. 8sin 4cos 3x x 3. sin 0.6x 4. 7cos 3 0x Find the exact functional value without using a calculator.

5. 1 2sin cos2

6. 1 4sin sin3

7. 1 1cos2

8. sec 2x 9. 1 3tan3

Solve each equation on the interval 0, 2 . Give exact values.

10. 22sin 1 0x 11. 2sin cos sinx x x 12. 24sin 4sin 1 0x x 13. tan sinx x 14. 2sec 1 tanx x 15. cos2 sin 1x x

16. cos 3cos 2x x 17. 22 tan 2 3sec 2x x 18. 10sin 2 5x Solve each equation for all real values of x (in Radian). Give exact values. 19. 22sin 5sin 2 0x x 20. 2sin 2sin 3 0x x 21. 3cos2 5cos 1x x 22. 2 tan cos 2cos tan 1x x x x

Math Analysis H WS 5.1- DeMoivre's Theorem Day 1 Calculate each given product and express your answer in the form a bi .

1. 6

cos sin12 12

i

2. 20

cos sin5 5

i

3. 5

7 73 cos sin30 30

i

4. 12

3 7 74 cos sin36 36

i

5. 121 i

6. 82 2i

7. 10

3 12 2

i

Find the nth roots of each given number in polar form. Pick 2 of the following.

8. 64 cos sin ; 35 5

i n

9. 8 cos sin ; 310 10

i n

10. 81 cos sin ; 412 12

i n

11. 16 cos sin ; 57 7

i n

Solve the given equation in the complex number system. Pick 2 of the following. 12. 3 27 0x i 13. 6 729 0x 14. 4 1 3x i 15. 4 8 8 3x i

Math Analysis H WS 5.2- DeMoivre's Theorem Day 2 Calculate each given product and express your answer in the form a bi . Use both Polar Form and DeMoivre's Theorem.

1. 20

1 32 2

i

2. 141 1

2 2i

3. 81 3i

Find all indicated roots of unity and express your answers in the form a + bi. 4. Fourth roots of unity 5. Sixth roots of unity Find the nth roots of each given number in polar form. 6. 1 ; 2i n Solve the given equation in the complex number system. 7. 6 64 0x 8. 3x i Represent the roots of unity graphically. Then use the trace feature to obtain approximations of the form a + bi for each root. Round to the nearest ten-thousandth. 9. Seventh roots of unity 10. Fifth roots of unity 11. Eighth roots of unity 12. Twelfth roots of unity 13. Ninth roots of unity

Math Analysis H Classwork- Graphing Investigation I: Changing Values of n Switch your calculator mode to POL (Polar Mode) and radians. Pressing Y= now takes you to a menu containing 1r . Further, your calculator will no

longer display an "X" when you press the button. Instead, it will display . So you will input an angle measurement and will get an output that is the length of a point to the origin. Thus, you are graphing the ordered pairs ,r . Finally, your window will be extremely important. In general, your window settings should be as follows..

Min: 0 X Min: -6 Y Min: -4 Max: 2 or 360 X Max: 6 Y Max: 4 Step: 0.03 or 3° X Scl: 1 Y Scl: 1

For different problems, however, you may wish to change the max/min on the two axes to better see the graph. For Problems 1-5 below...

Graph the equation using at least 4 different values of a for each problem. Values should be 4 4a .

On a separate sheet of paper: Record the general form. Describe its basic shape. Describe exactly what the value of a does to the graph. Include in

your description what happens when a is positive versus negative. Write the rectangular form of each polar equation in terms of a.

1. General Form: r a 2. General Form: sinr a 3. General Form: cosr a 4. General Form: cscr a 5. General Form: secr a

6. The calculator cannot graph a . What would the graphs of 3 or

4 look like? Sketch these two graphs. Then describe the graph of

the general form of a .

Math Analysis H Classwork- Graphing Investigation II: Polar Curves Investigation Use the same calculator settings as Graphing Investigation I The Rose General Form: sinr a b or cosr a b

Graph at least three different equations for sin and at least four different equations for cos. Be sure to choose both positive and negative values for a ( 4 4a ) as well as both even and odd values for b. Next to each graph, describe the basic shape, describe how the values of a and b that you chose affect that graph, and describe the difference between using sin and cos.

The Leminscate

General Form: 2 2 cos 2r a or 2 2 sin 2r a NOTE: You must take the

square root of both sides since it is 2r . Graph at least three different equations, changing the value of a as well as changing between sin and cos. Next to each graph, describe the basic shape, describe how the value of a that you chose affects that graph, and describe the difference between using sin and cos.

The Limacon General Form: cosr a b or sinr a b

Graph at least three different equations, changing both the value of a and b as well as changing between sin and cos. Next to each graph, describe the basic shape, describe how the values of a and b that you chose affect that graph, and describe the difference between using sin and cos.

a. What is/are the requirements to have a loop in the middle versus just

an indent? b. For 3cosr a , for what value(s) of a will the graph have neither

a loop nor an indentation? c. What is the relationship between a and b so that there is neither a loop

nor an indentation? d. Create a cosine equation where there is a loop in the middle. e. Create a sine equation where there is only an indentation.

The Cardioid (A Special Limacon) General Form: cosr a a or sinr a a

Graph at least three different equations, changing the value of a as well as changing between sin and cos. Next to each graph, describe the basic shape, describe how the value of a that you chose affects that graph, and describe the difference between using sin and cos.

a. Why do you suppose we call this graph a Cardoid?

Math Analysis H WS Polar- Transitions 1. Consider the graph at the right.

a. What rectangular equation could be used to represent the graph?

b. What Polar Equation could be

used to represent the graph? 2. It's helpful to remember that

2 2 2x y r . This is a link between the two equation forms. Consider the diagram to the right.

Using this diagram, identify three basic equations useful in converting rectangular equations to polar form. Substitution is a useful tool in this process.

3.

a. Graph the circle with the rectangular equation 22 4 16x y ? b. By substituting basic identities from question 2, convert the equation to

a polar equation and solve for the variable r. c. Graph the polar equation on your calculator.

4. Graph 2

cosr

on your calculator.

a. Describe the graph. b. If you were told this was the graph of a rectangular equation, what

would you determine the equation to be? c. Check your answer by converting the equation from polar to

rectangular. 5. Write the polar form of each of the following equations

a. 2 2 64x y b. 2 22 4x y

c. 5x d. 2 2 1x y

6. Write the rectangular form of each of the following polar equations:

a. 3r b. 3sinr

c. 6 cos4

r

d. 3sec 60r

7. Write the polar from of each of the following linear equations: (Graph to

check your answers).

a. 0x b. 0y c. 3x

d. 2y e. y x f. 2y x

g. 13

y h. 4 1y x i. 1 62

y x

Math Analysis H WS Polar- Systems of Polar Equations Sketch a graph of each system. Determine all solutions algebraically (show your work), then determine all points of intersection. Label intersections on your graph.

a. 32cos

rr

b. sin1 sin

rr

c. 2sin

2 3 cos

r

r

d. 2sin2cos2

rr

e. 2sin2sin2

rr

f. 2sec

rr

g. 12cos2

rr

Math Analysis H WS Polar- Enrichment

Distance Using Polar Coordinates Suppose you were given the polar coordinates of two points 1 1 1,P r and 2 2 2,P r and were asked to find the distance, d, between the points. One way would be to convert to rectangular coordinates 1 1,x y and

2 2,x y and apply the distance

formula: 2 22 1 2 1d x x y y

A more straightforward method makes use of the Law of Cosines. 1. In the above figure, the distance d between 1P and 2P is the length of one

side of 1 2OP P . Find the lengths of the other two sides. 2. Determine the measure of 1 2POP . 3. Write an expression for 2d using the Law of Cosines. 4. Write a formula for the distance, d, between the 1 1 1,P r and 2 2 2,P r . 5. Find the distance between the points (3, 45°) and (5, 25°). Round your

answer to three decimal places.

6. Find the distance between the points 2,2

and 4,8

. Round your

answer to four decimal places. 7. The distance from the point (5, 80°) to the point (r, 20°) is 21 . Find r.

Math Analysis H WS Vectors- Day 1

Find the magnitude of vector PQ

. 1. 2, 3 , 5, 9P Q 2. 3, 5 , 7,11P Q 3. 7, 0 , 4, 5P Q 4. 30, 12 , 25, 5P Q

Find a vector equivalent to the vector PQ with its initial point at the origin.

5. 1, 5 , 7,11P Q 6. 2, 7 , 2, 9P Q 7. 4, 8 , 10, 2P Q 8. 5, 6 , 7, 9P Q

9. 14 17 12, 2 , ,5 5 5

P Q

10. 2, 4 , 3, 1P Q

For each of the following, find u v , u v , and 3 2u v .

11. 2, 4 , 6, 1u v 12. 4, 0 , 1, 3u v

13. 3, 3 2 , 4 2,1u v

14. 2 19, 4 , 7,3 3

u v

15. 12 2, 5 , 7,124

u v

Given 3, 1 , 8, 4 , and 6, 2u v w , find the magnitude of each vector.

16. u v 17. u v 18. 3u v 19. v w 20. 2 v w 21. 2 2w u 22. 0.5u w

Math Analysis H WS Vectors- Law of Cosines Complete the following, neatly, on a separate sheet of paper. Include a diagram for each problem. 1. A hiker leaves camp and walks 13 km due north. The hiker then walks 15

km northeast (exactly midway between north and east). Find the hiker’s direction and displacement from her starting point.

2. A pilot flies a plane east for 200 km, then 60° south of east for 80 km. Find

the plane’s distance and direction from the starting point. 3. An airplane flies due west at 185 km/h with respect to the air. There is a

wind blowing at 85 km/h to the northeast. What is the plane’s speed and direction from the starting point?

4. An airplane travels on a bearing of 100° at an airspeed of 190 km/h while a

wind is blowing 48 km/hr toward a bearing of 220° find the ground speed and direction of the plane from its starting point.

5. A ship leaves port at 1 pm traveling north at the speed of 30 miles/hour. At

3 pm, the ship adjusts its course 20 degrees eastward. How far, and at what bearing, is the ship from the port at 4pm?

Math Analysis H WS Vectors- Vector Basics What is the magnitude and direction of the following? 1. 2. 3. Sketch the resultant vector of each of the following. 4. 5. 6. What is the magnitude and direction of the resultant in the sketch below? 7. Choose any three of problems 8-17 to complete. 8. Vector u

has a magnitude of 20 and a direction of 0°. Vector v

has a

magnitude of 40 and a direction of 60°. Find the magnitude and direction of the resultant to the nearest whole number.

9. Vector u


has a


10. Vector u


has a


11. Two forces with magnitudes of 20 pounds and 14 pounds and an angle of 55° between them are applied to an object. Find the magnitude of the resultant vector to the nearest whole number.

12. Two forces with magnitudes of 48 pounds and 65 pounds and an angle of

80° between them are applied to an object. Find the magnitude of the resultant vector to the nearest whole number.







16. Vector u


has a


17. Vector u


has a


18. Use vectors to answer the following questions.

a. If two forces from different directions are applied to an object, can the magnitude of the resultant be larger than the sum of the magnitudes of the forces? Justify your answer.

b. What if the forces were from the same direction? Justify your answer.

28°

8.5 lb 110° 4.5 lb 70° 12 lb

20° 115° 85°

26°

48°

116° 10

13 6

Math Analysis H WS Vectors- Word Problems Complete the following, neatly, on a separate sheet of paper. Include a diagram for each problem. 1. Two ropes are tied to a wagon.

A child pulls one with force of 20 pounds, while another child pulls the other with a force of 30 pounds. If the angle between the two ropes is 28°, how much force must be exerted by a third child, standing behind the wagon, to keep the wagon from moving? (Hint: Assume the wagon is at the origin and one rope runs along the positive x-axis. The third child must use the same amount in the opposite direction.)

2. Two circus elephants, Bessie and

Maybelle, are dragging a large wagon. If Bessie pulls with a force of 2200 pounds and Maybelle with a force of 1500 pounds and the wagon moves along the dashed line, what is angle ?

3. For an airplane with given air speed 250 miles per hour in the direction of

60° and wind speed of 40 miles per hour from the direction 330°, what is the speed and direction of the actual flight of the plane?

4. A plane is flying in the direction 200° with an air speed of 500 miles per

hour. Its course and ground speed are 210° and 450 miles° per hour, respectively. What is the direction and speed of the wind?

5. A river flows from east to west. A swimmer on the south bank wants to

swim to a point on the opposite shore directly north of her starting point. She can swim at 2.8 miles per hour, and there is a 1- mile-per-hour current in the river. In what direction should she swim in order to travel directly north (that is, what angle should the swimmer make with south bank of the river)?

6. A river flows from west to east. A swimmer on the north bank swims 3.1

miles per hour along a line that makes a 75° angle with the north bank of the river and reaches the south bank at a point directly south of his starting point. How fast is the current in the river?

Math Analysis H WS Sequences and Series- Sequences and Series Review Complete the following, neatly, on a separate sheet of paper. Show all formulas used and work. 1. If 1 5a and 10d , find the ninth term of the arithmetic sequence. 2. Given the arithmetic sequence -4, 1, 6, ... and 61na , find n . 3. Find the missing terms of the arithmetic sequence 6, __, __, __,72. 4. If 1 10a , 2d , and 12n , find the sum of the arithmetic series. 5. If 3d , 7n , and 16na , find the sum of the arithmetic series. 6. If 1 3a , 30na , and 214.5nS , find the first three terms of the

arithmetic series. 7. A clock chimes each hour equal to the hour that just occurred (once for

1:00, twice for 2:00, etc). Find the number of chimes it makes in 24 hours. 8. If 1 3a and 2r , find the seventh term of the geometric sequence. 9. Find the missing geometric means for the sequence 3, __, __, __, __, -96. 10. If 1 32a , 0.5r , and 5n , find the sum of the geometric series. 11. If 2 2.2a , 5 17.6a , and 6n , find the sum of the geometric series. 12. If 5n , 3r , and 242nS , find the first term of the geometric series. 13. Find the sum of the infinite geometric series 7, 3, 9/7, 27/49, ... 14. Write a recursive formula of the sequence 1, 3, 7, 15, ... 15. Write a formula for the nth term of the sequence 200, 20, 2, 0.2, ...

16. Write 3

14 3

tt

in expanded form.

17. Find the sum of 6

3

13b

t

18. Use sigma notation to express 3 + 5 + 7 + ... + 21. 19. A certain ball dropped from a distance of 20 feet rebounds 3/4 of the

height from which it fell. Find the distance it travels before coming to rest.

Math Analysis H WS Sequences and Series- Binomial Theorem Partner Practice Complete neatly on a separate sheet of paper. This is due at the end of the period. 1. Write out the Binomial Theorem 2. Expand 53w x

3. Expand 45y

4. Find the 7th term of the expansion of 102 33x y

5. Consider the expansion of 922x y

a. Which term contains 5 8x y ? b. What is the coefficient of that term?

6. Find the coefficient of the term containing 6 5x y in the expansion of

82 3x y

-------------------------------------------------------------------------------------------------------- Math Analysis H WS Sequences and Series- Induction Partner Practice Prove the following with math induction. Show all steps and work as demonstrated in class. This is due at the end of the period.

1. 1 21 2 2 3 3 4 ... 1

3n n n

n n

2. 1 1 2

1 3 6 ...2 6

n n n n n

3. 3 11 4 7 ... 3 2

2n n

n

4. 0 1 2 3 11 2 2 2 3 2 4 2 ... 2 1 1 2n nn n

Math Analysis H WS Sequences and Series- Binomial Expansion and Math Induction Review Complete the following without the use of a calculator. Evaluate.

1. 12!9!

2. 10! 8!6!

3.

7 7 80 5 3

Expand the binomial and express the result in simplified form.

4. 32 1x 5. 42 1x 6. 52x y 7. 6x a

Write the first three terms of the binomial expansion.

8. 82 3x 9. 93x

Find the indicated term of the binomial expansion. 10. 52x ; 4th Term 11. 62 3x ; 5th Term Use mathematical induction to prove each statement is true for every positive integer, n.

12. 5 1

5 10 15 ... 52

n nn

13. 2 1 4 11 4 4 ... 43

nn

14. 22 6 10 ... 4 2 2n n

Also do page 865 1-31 odd & 32-36

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