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Geometry Assignments: Trigonometry

|Day |Topics |Homework |HW Grade |Quiz Grade |

|1 |Intro to trig |HW Trig - 1 | | |

|2 |Trigonometric ratios |HW Trig – 2 | | |

|3 |Inverse Trigonometric Functions |HW Trig - 3 | | |

|4 |Solving triangles |HW Trig - 4 | | |

|5 |Word problems |HW Trig - 5 | | |

|6 |Special angles and cofunctions |HW Trig - 6 | | |

|7 |Practice |HW Trig - 7 | | |

|8 |Harder Problems |HW Trig - 8 | | |

|9 |Review Day |HW Trig - Review | | |

| |**TEST** | | | |

NOTE: Make sure your calculator is in DEGREE mode before using sin, cos, tan, sin-1, cos-1 or tan-1.

MODE

((( (Highlight “degree” on line 3.)

ENTER

Answers to selected problems

HW - 1

1. 16.4 2. 34.8 3. 10.2 4. 31.8 5. 6.8 6. 31.3 7. 61(

8. 27( 9. 37( 10b. sinA = cosB and cosA = sinB 11c. They are recprocals

HW - 2

1a. = 0.75 b. [pic]( 0.6614 c. [pic]( 1.1339 d. [pic] ( 0.6614

e. = 0.75 f. ( 0.8819

2a. Increases b. Decreases c. Increases 3a. FALSE b. FALSE c. FALSE

4a. sin Q, cos P b. cos Q, sin P c. tan Q

5a. [pic] b. [pic] c. [pic] d. 1

6a. AA b. 12.5 c. tan A = tans P = 0.625 d. 0.625

7a. Yes since 82 + 152 = 172 b. 15/17 8a. Yes since it’s similar to Jim’s (. b. 15/17

9. No; 152 + 252 ( 282 b. 15/17 10a. 7.482 b. 22.7

11. Draw an altitude in an equilateral triangle to form two 30(-60(-90( triangles; erase/cross out one of them. Choose appropriate values for the lengths of the hypotenuse and the shorter leg (think about the relationship between them). Use Pythagoras to find the length of the longer leg. Then read sin 60( off the triangle (the answer, in simplest form, is [pic]).

12a. Yes, it’s correct. To solve an equation, we use opposite (or inverse) operations. The opposite of multiplication by 3 is division by 3.

b. Only a doofus would do this. First of all, [pic], without a number inside it, is not a number and you can’t divide by it. (It would be a bit like trying to divide by a plus sign; it makes no sense.) Think, what is the opposite (inverse) of “square root”?

HW - 3

1. 60° 2. 45° 3. 20° 4. 19° 5. 30° 6. 72° 7. 14.41

8. 20° 9. 21.46 10. 85° 11. 67° 12. 35.49 14a. 37° b. 2.29

HW - 4

1. 23.32 2. 27° 3. 12.86 4. 56.61 5. 19° 6. 41° 7. 22.07

8. x = 13, y ( 34( 9. x = 3, y = 30( 10b. (72( 11a. 31.07 b. 44.46 12. Molly. Why?

HW - 5

1. 64° 2. 1327 3. 31.2 4. 121' 5a. 10.8( b. 6.2' 6. 5344' and 5115'

7. 130'

HW - 6

1. 45(, 8, [pic] 2. 30(, 9, [pic] 2a. 65 b. 22.5 c. 40 d. 8 e. 0 ( θ ( 90( [pic]

3&ha. [pic] b&g. [pic] c. [pic] d&e. [pic] f. 1 i. [pic]

4c. 63( 5d. 34( 6a. 40' b. 30' 7. About 475 mph

HW – 7

1a. at least 8.84' b. 34( 2. 110.4' 3. Either 14.7' or 53.3' 4. 47 5. 2214.6'

6a. 10' b. 60(

HW – 8

1. 324 m 2. 1111 feet 3. 38.3(; 30.4 4. 136.8 feet

Geometry HW Trigonometry - 1

Name

Use the table in the notes to do the following. Show work.

In 1 – 6, find the missing side.

1. 2. 3.

4. 5. 6.

In 7 – 9, find the missing angle.

7. 8. 9.

10. a. Draw (ABC with right angle C. Choose any two

different values to be the lengths of the legs. Find

the length of the hypotenuse (leave in radical form

if appropriate).

b. Find the value of opp/hyp, adj/hyp and opp/adj for angle A.

A: opp/hyp = adj/hyp = opp/adj =

Find the same ratios for angle B.

B: opp/hyp = adj/hyp = opp/adj =

How do the ratios for angle A compare to the ratios for angle B?

11. a. Pick an angle in the table from 1 to 89 degrees. Write down the decimal approximations for the ratios opp/hyp and adj/hyp.

Angle = opp/hyp = adj/hyp =

b. Find the complement of your angle. Write down the decimal approximations for the ratios opp/hyp and adj/hyp. How do they compare to the answers from part a?

Complement = opp/hyp = adj/hyp =

c. Find the product of the ratio opp/adj for each of your two angles in a and b (round to the nearest thousandth). What does this tell you about the two ratios?

12. You MUST know the terminology (opposite, adjacent and hypotenuse) in the diagram at right. (You should already know most of it from the notes and be able to figure out the rest using common sense.)

Geometry HW Trigonometry – 2

Name

Note: Most of these problems are not “Do exactly what we did in class” type exercises. You should think about them and learn from them. You are not trying simply to get the answers, you need to understand them. Have fun.

1. Use the diagram to find the numerical value of each trig function

both as a fraction and as a decimal to the nearest ten-thousandth:

a. sin A = b. cos A = c. tan A =

d. sin B = e. cos B = f. tan B =

2. Experiment with your calculator to figure out the following. As the degree measure of angle A increases from 0° to 90°, determine whether the value of each of the following increases or decreases:

a. sin A increases/decreases b. cos A increases/decreases c. tan A increases/decreases

3. Experiment some more on your calculator to decide if the following are True or False.

Important note: sin 2A means sin (2A); not (sin 2)A.

a. sin 2A = 2sin A b. cos 2A = 2cos A c. tan 2A = 2tan A

4. In right triangle PQR, PR = 24, QR = 7 and hypotenuse PQ = 25.

From that triangle:

a. Name two trig functions that have the value .

Note: to be meaningful, trig functions need an angle.

“tanA” means “tangent of A;” “tan” just means “light brown.”

b. Name two trig functions that have the value 0.28.

c. Name one trig function that has the approximate value 3.4286.

5. In (RST, R is a right angle and cos T = . Draw a possible diagram of

(RST and find the values of

a. sin S = b. cos S = c. tan S =

d. [pic] Important note: [pic] means [pic] , not [pic] or [pic] .

6. In the triangles at right, (A ( (P.

a. Explain why the two triangles are similar.

b. Find the value of h.

c. Find tan A and tan P in decimal form.

tan A = tan P =

d. Suppose a third right triangle contains (K

with (K ( (A. Find tan K.

7. Tim has a triangle with sides measuring 8, 15, and 17.

a. Is Tim’s triangle a right triangle? Justify your answer.

b. If T is the larger acute angle in Tim’s triangle, find the value of sin T as a fraction in simplest form.

8. Kim has a triangle that is similar to Tim’s (see problem 8 above) but is exactly 692,317.4 times as large.

a. Is Kim’s triangle a right triangle? Justify your answer.

b. If K is the larger acute angle in Kim’s triangle, find the value of sin K as a fraction in simplest form.

9. Jim has a triangle with sides 17, 25 and 28. (J is between the sides measuring 17 and 28.

a. Is Jim’s triangle a right triangle? Justify your answer.

b. Jim’s older brother Slim, who takes Alg2, uses the Law of Cosines to show that (J is congruent to (T in Tim’s triangle (problem 8 above). Find the value of sin J as a fraction in simplest form.

10. Solve the following equations for x. Express answers to the nearest tenth.

a. 0.6235 = [pic] b. 0.3528 = [pic]

11. MacGyver is stranded on a desert island without his calculator. To get off the island, he is building a hang glider out of coconut shells and palm leaves. As part of the design, he needs to know sin 60(. Explain, concisely but completely, how MacGyver can find the answer to his problem. (Hint: Start with an equilateral triangle.)

12. a. Rufus has to solve the equation 3x = 36. He decides to divide both sides by 3. Is this the correct thing to do? Explain why. (Note: “That’s just how you do it” or “That’s just not how you do it” are not explanations. Neither is “Mr. LaShomb said so.” Think of a more mathematical explanation.)

b. Doofus has to solve the equation [pic]. He decides to divide both sides by [pic]. Is this the correct thing to do? Explain why.

Geometry HW Trigonometry - 3

Name

Use your calculator to find the value of x to the nearest degree for each of the following. No work is required for #1 – 6, just answers.

1. sin x = .8660 2. cos x = .7071 3. tan x = .3640

4. sin x = 5. cos x = 6. tan x = 3

Solve the following equations. Just an answer is not enough. Show what you put into your calculator to

find x. For example, [pic] or [pic]. Round all angles to the nearest degree. Round all other variables to the nearest hundredth.

7. tan 42° = [pic] 8. sin x = 9. cos 56° = [pic]

10. tan x = 11. cos x = 12. sin 25° = [pic]

13. Rufus had to solve cos x = 0.8. He used his calculator and got x = 0.9999 which he rounded to 1. His teacher, Mr. Hipparchus, marked him wrong. What was Rufus’s mistake and what was the correct answer?

14. Find the value of x in each diagram below. Show work.

a b.

Geometry HW Trigonometry - 4

Name

Use the information in each figure to write an equation, solve the equation to get x by itself, and then find the value of x. Round angles to the nearest degree and sides to the nearest hundredth.

For problems #8 and 9, first find the value of x. Then find m(A.

8. 9.

10. Triangle OMG has vertices O(0, 0), M(9, 3) and G(10, 0).

a. Prove using coordinate geometry that (OMG

is a right triangle.

b. Find m(G.

11. An isosceles triangle has a base of length 12 and base angles measuring 51(.

a. Find the perimeter of the triangle to the nearest hundredth. (Note that you have a problem: You only know how to do trigonometry with right triangles. Figure out a way to fix this problem.)

b. Find the area of the triangle to the nearest tenth.

12. The following question appeared on one of Mr. Hipparchus’s math quizzes:

Using the diagram at right, find the value of sin A.

Dolly wrote [pic].

Holly wrote [pic].

Molly wrote [pic].

Polly wrote 41(.

Who got full credit and why? (Try not to mess this up on a quiz.)

Geometry HW Trigonometry - 5

Name

For each problem, draw an appropriate diagram, write an equation, solve the equation to get x by itself and then find the value of x.

1. A guy wire (a wire, not necessarily male, used to stabilize something) 115' long runs from a radio tower to a point on the ground 50' from the center of the tower's base. What angle does the wire make with the ground?

2. A tourist on the South Rim of the Grand Canyon observes a point on the North Rim at an angle of elevation of 1.2(. The canyon is 12 miles wide at that point. How many feet higher is the North Rim than the South Rim there?

3. Given that A, P and B are collinear, find the length of [pic] in

the diagram at right to the nearest tenth.

4. A boy flying a kite lets out 150' of string that makes an angle of 52° with the horizontal. The boy holds the end of the string at a height 3' above the ground. Assuming the string is stretched taut, find to the nearest foot how high the kite is above the ground.

5. Ralph is loading some stuff onto a truck with the help of a handcart. When he uses 10-foot long boards to make a ramp up to the back of his truck, the boards make an angle of 17.5( with the ground. This turns out to be too steep.

a. What will be the angle if he uses 16' long boards?

b. How much farther behind the truck will the 16' boards hit the ground compared to the 10' boards?

6. The map shows the three tallest mountain peaks in New York State: Mount Marcy, Algonquin Peak, and Mount Haystack. Mount Haystack, the shortest peak, is 4960 feet tall. Surveyors have determined the horizontal distance between Mount Haystack and Mount Marcy is 6336 feet and the horizontal distance between Mount Marcy and Algonquin Peak is 20,493 feet. The angle of depression from the peak of Mount Marcy to the peak of Mount Haystack is 3.47 degrees. The angle of elevation from the peak of Algonquin Peak to the peak of Mount Marcy is 0.64 degrees. What are the heights, to the nearest foot, of Mount Marcy and Algonquin Peak? Justify your answer.

7. Rufus wants to rig a zip line from the flat roof of a tall building to the flat roof of a shorter nearby building. From the roof of the shorter building, 30 feet above the ground, the angle of depression to the base of the taller building is 14.036( and the angle of elevation to the roof of the taller building is 22.620(. Find the length of the zip line. (It’s helpful to first find the distance between the buildings.)

Geometry HW Trigonometry - 6

Name

1. In each diagram, find the values of a, x and y. Irrational answers may be left in radical form. In both problems, a represents the measure of an angle; x and y represent the measures of sides,

a. b.

2. Find a value of θ (in degrees) that will solve each equation.

a. [pic] b. [pic] c. [pic]

d. [pic] 3. [pic]

3. Find the exact value of each of the following without using your calculator.

a. sin 45( = b. cos45( = c. tan 45( =

d. sin 30( = e. cos 30( = f. tan30( =

g. sin60( = h. cos 60( = i. tan 60( =

4. Triangle ABC has vertices A(3, 6), B(4, –2), and C(0, 4).

a. Show using coordinate geometry that (C is a right angle.

b. Find the lengths of [pic]and [pic].

c. Find the measure of (A to the nearest degree.

5. a. Graph the line 2x – 3y = 6.

b. What is the slope of the line in lowest terms?

c. What is the tangent of the angle the line makes with the x-

axis in lowest terms?

d. To the nearest degree, what angle does the line make with the x-axis?

6. Doofus has a tree house 17 feet off the ground. His neighbor Goofus has an even higher tree house. From Doofus's tree house, the angle of elevation to Goofus's tree house is 18°. The angle of depression to the bottom of the tree containing Goofus's tree house is 23°.

a. How far apart are Doofus's and Goofus's trees to the nearest foot?

b. How high is Goofus's tree house to the nearest foot?

7. An airliner flies over the Pacific toward Isla Nublar. It flies at a constant altitude of 38,000 feet. A radar station on the island is 6,000 feet above sea level. At 10:25 AM, the angle of elevation from the radar to the aircraft was 3.5(. At 10:28 AM, the angle had increased by 1.1(. How fast is the aircraft flying to the nearest mile per hour?

Geometry HW Trigonometry - 7

Name

1. Orville Gunkmeuller’s 12 foot wide one-car garage sticks

out to the side of his house. He has a twenty foot length of

single strand electric fence running from the far corner of

the garage to the back corner of the house forming a

triangular area to keep his goats in. (See diagram at right.)

After winning the lottery, Orv buys a new car, boat, snow-

mobile and four-wheeler and decides to double the width of his garage.

a. How much additional wire will Orv need to fence in the new triangle formed by the house and garage?

b. To the nearest degree, what angle will the new fence make with the side of the house?

2. A climber is scaling a vertical cliff face. Her boyfriend is watching her through binoculars while sitting on the ground 100' feet from the base of the cliff. From where he sits, the angle of elevation to the climber is 35° and the angle of elevation to the top of the cliff is 61°. How far is the climber from the top of the cliff?

3. Logan rides a Ferris wheel having a radius of 30 feet and with its lowest point 4 feet above the ground. Let L represent Logan and C represent the center of the Ferris wheel. When the wheel stops to let passengers off and on, the angle between [pic] and the horizontal is 40(. How high is Logan above the ground?

4. A regular pentagon is inscribed in a circle of radius 8. Find the perimeter of the pentagon.

(Suggestion: Draw radii to two consecutive vertices of the pentagon, find the

measure of the vertex angle of the triangle, then drop an altitude.)

5. A sailboat sails toward a lighthouse atop a steep cliff (see diagram, not even close to scale). The angle of depression from the deck of the lighthouse to the top of the mast of the sailboat is 3.3(. To the nearest foot, how far is the bow (front) of the sailboat from the base of the cliff?

6. A lock on the Dreary Canal is 18 feet wide. When closed, the gates form an angle of 128(. The gates part in the middle and swing open to let boats into and out of the lock.

a. How wide, w, is each lock gate?

b. What minimum angle x (see diagram) must the gates make to create an opening that a boat 8 feet wide could fit through?

Geometry HW Trigonometry - 8

Name

1. Brigitte (remember her?) is trying again to find the height of the Eiffel tower. She measures

the angle of elevation from her position to the top of the Eiffel tower to be 20.22(.

She then walks 100 meters directly toward the tower and measures the angle

of elevation again; this time it is 22.58(. Assuming Brigitte measured her angles from eye level, 172 cm above

the ground, find the height of the Eiffel

tower to the nearest meter.

2. A surveyor in Wyoming uses a theodolite (a device for measuring angles)

on a five foot tall tripod to find that the angle of elevation from his

location to the top of a tall butte is 11.5°. He walks over level

ground 1000' directly toward the butte at which point the

angle of elevation to the top of the butte is 14°. How high is the butte?

3. Two right circular cones both have height 24. The larger cone has a diameter that is 25 more than the diameter of the other. The angle the side of the narrower cone makes with the horizontal is 15( more than the angle, a, the side of the larger cone makes with the horizontal. Find the angle the side of larger cone makes with the horizontal and find the radius of that cone. (Note: ZoomFit will probably not give you a fantastic window. After you look at it, go back to WINDOW, adjust Ymax appropriately, and then hit GRAPH. It may take more than one try.)

4. Winifred Wellesley wants to find the length of Paramecium Pond. She marks points at the east (E) and west (W) ends of the pond. From a third point P, the distance to W is 121 feet and the distance to E is 85 feet. The angle WPE measures 81.2026( (Winifred measures angles extremely accurately).

a. You need a right triangle. You will draw an altitude. Explain why the altitude from P is not going to be helpful. (An altitude from either E or W will do.)

b. How long is the pond?

Geometry HW Trigonometry - Review

Name

1. In right triangle ABC, if m(C = 90 and cos A = 8/17, what is tan B?

2. Find a value of A for which [pic].

3. Solve for x in each figure.

a. b. c.

4. In (ABC, (C is a right angle. If BC = a, AC = b, and AB = c,

a. What trig function has the same value as sin A?

b. What trig function has the same value as cos A?

c. How are the values of tan A and tan B related?

5. a. The intermediate ski slope at Gory Mountain is angled 24( from the horizontal. If Suzy Slalom skis 120

yards straight down the slope, what horizontal distance has she traveled?

b. While skiing 180 yards down the expert slope at Gory Mountain, Suzy drops a vertical distance of 120

yards. What (average) angle does the expert slope make with the horizontal?

6. The extension ladder on a fire truck is extended to a total length of 32'. The base of the ladder is on top of the truck and about 8 feet off the ground. What angle does the ladder have to make with the horizontal to reach the bottom of a window 36' above the ground?

7. (This problem is in honor of Jaime Wilson, who made it up.) In the diagram, (RST is isosceles with (R ( (S and altitude [pic]. If RT = 4x – 7, ST = 2x + 3 and m(S = 50°, find

a. the length of [pic]and

b. the length of the base [pic].

8. In trapezoid RGHT, R and G are right angles, m(H = 119°, RG = 8, and RT = 15.

a. Find the area of RGHT to the nearest tenth.

b. Find the perimeter of RGHT to the nearest tenth.

c. Find the length of diagonal [pic]to the nearest tenth.

d. Find to the nearest degree the measure of (HRT.

9. A small aircraft is 8 miles horizontally from its intended runway and flying 5100 feet above the ground. For a normal landing, the pilot wants to begin his final approach at an altitude of 500’ and maintain an angle of descent of 3(. What angle of descent must the plane fly from its present position to the beginning of its final approach? In the diagram, the aircraft starts at A, final approach begins at F, and the runway is at R.

10. A drone flying at a height of 100 meters and a speed of 9 meters per second is sent out to search for the Indominus Rex.

a. When the drone first spots the Rex, the angle of depression from the drone to the dino’s head, about 3 meters off the ground, is 15(. If the drone immediately changes course to fly directly over the beast, how many seconds will it take the drone to be directly above the dinosaur?

b. When fully zoomed in, the camera on the drone has a field of view of 8(. If the Indominus Rex is 15 meters long, will the camera be able to capture the whole dinosaur in a single shot from 100 meters above it?

11. Dewitt Wright is building a small cabin by a lake. The cabin will be rectangular with exterior dimensions 18 feet by 30 feet. Local codes require a roof pitch (angle up from the horizontal) of at least 26.6( and the roof must overhang the cabin by 1 foot horizontally on all sides. If a bundle of shingles costs 27.95 and covers 33⅓ square feet, how much will he pay (excluding tax) for the shingles?

12. Avis Bird is sitting at the end of the crosspiece on a utility pole, 3.5 feet from the actual pole. Felis Cat is climbing the pole with dinner on his mind. Meanwhile, Canis Dog observes both of them from the ground 40 feet from the base of the pole. The angle of elevation from Canis to Felis is 32( while the angle of elevation from Canis to Avis is 43(. How far must Felis travel to get to Avis? (Note: Felis climbs well but flies poorly.)

Review Answers

1. 8/15 2. 32.5 3a. 9.1 b. 68( c. 13.6

4a. cos B b. sin B c. They are reciprocals. 5a. 109.6 yds b. 42(

6. 61 7a. 9.96 b. 16.7 8a. 102.3 b. 42.7 c. 13.3 d. 37°

9a. 8( 10a. About 40 (if Rex stands still) b. Not quite, Rex is a meter too long.

11. $614.90 12. Just over 12.5 feet.

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