Day 1 Notes/Practice

1. For an acute angle A in right triangle ABC, the trigonometric functions are as follow:

sin A = side opposite hypotenuse

cos A = side adjacent hypotenuse

tan A = side opposite side adjacent

2. A right triangle has sides whose lengths are 8 cm, 15 cm, and 17 cm. Find the values of missing angles.

3. In triangle RST, find the measure of R to the nearest tenth.

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148

RS

T

B

C A

4. Solve each right triangle. Round angles to the nearest tenth and sides to the nearest tenth.

To SOLVE A TRIANGLE, means to __________________________________________

a. C=55 ° ,c=16 b. A = 49°, a = 10

c. A=41 ° , b=7.44 d. A=37 ° ,b=11

e. A=45 ° , c=7√2 f. A = 20°, c = 35

5. Angle of Elevation

6. Angle of Depression

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7. From a point on level ground 125 feet from the base of a tower, the angle of elevation is 57.2 ° . Approximate the height of the tower to the nearest foot.

8. A kite flies at a height of 30 feet when 65 feet of string is out. If the string is in a straight line, find the angle that it makes with the ground. Round to the nearest degree.

9. You are standing on level ground 800 feet from Mt. Rushmore, looking at the sculpture of Abraham Lincoln’s face. The angle of elevation to the bottom of the sculpture is 32 ° and the angle of elevation to the top is 35 °. Find the height of the sculpture of Lincoln’s face to the nearest tenth of a foot.

Solve each right triangle. Round lengths to tenths and angles to given units.

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10. Mrs. James is using a 6-meter ladder to clean the windows on her second floor. Her ladder stands on level ground and rests against the side of her house at a point 4 meters from the ground. How far from the side of her house is the foot of the ladder? Round your answer to the nearest hundredth.

11. The Washington Monument is 169.29 meters tall and, at a particular time, casts a shadow 201.2 meters long. Find the approximate angle of elevation of the sun at the time. (nearest degree)

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Day 1 Homework – Word Problems Draw the right triangle and solve the problem.

Round all lengths to tenths and angles to nearest degree.

1. A 13.5 meter ladder is leaning against a wall. Find the distance the ladder goes up the wall if it makes an angle of 43 ° with the ground.

2. A guy wire that is 77.4 meters long is attached to the top of an antenna mast that is 71.3 meters high. Find the angle that the wire makes with the ground.

3. Find the length of a guy wire that makes an angle of 45 ° with the ground if the wire is attached to the top of the tower 63 meters high.

4. Suppose the angle of elevation of the sun is 23.4 °. Find the length of the shadow cast by Cindy Newman, who is 5.75 feet tall.

5. The shadow of a vertical tower is 40.6 meters long when the angle of elevation of the sun is 34.6 °. Find the height of the tower.

6. Find the angle of elevation of the sun if a 48.6 ft flagpole casts a shadow 63.1 ft long.

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7. The angle of depression from the top of a building to a point on the ground is 32 °. How far is the point on the ground from the top of the building if the building is 252 meters high?

8. An airplane is flying 10,500 feet above the level ground. The angle of depression from the plane to the base of a tree is 13 ° . How far must the plane fly horizontally to be directly over the tree.

9. The angle of elevation from the top of a small building to the top of a nearby taller building is 46 °, while the angle of depression to the bottom of the taller building is 1 4 ° . If the smaller building is 28 meters high, find the height of the taller building.

10. Shelly McCarthy knows that when she stands 123 feet from the base of a flagpole, the angle of elevation is 26 ° . If her eyes are 5.3 feet above ground, find the height of the flagpole.

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Recall: AA, SAS, and SSS Similarity

For Practice:

Problem 1 Problem 2 Problem 3

∆CBA ∆ ¿ ∆ FSR ∆¿ ∆ STU ∆¿

Similar by ______ Similar by ______ Similar by ______

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Day 2 Law of Sines

*Used when not a right triangle and you have SSS, ASA, and AAS from geometry

***Make sure you are in _____________ mode!!!!

Important notes:

*Side lengths are represented with __________________________________

*Angle measures are represented with _______________________________

*The largest side is opposite the _________________ angle

1. Find c if B = 101˚, C = 63˚, and a = 9

2. Find C is A = 88˚, a = 26, c = 21

3. Solve triangle ABC if A=32°, B=57 °, and c=14.3. Round angle measures and side measures to the nearest tenth.

4. Practice: Solve triangle ABC if B = 76˚, C = 60˚, and b = 8.

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The Ambiguous Case (SSA)SSA does not guarantee a unique triangle (or even that a exists). Therefore SSA is called the

__________________ case.

SSA means 2 sides and a non-included angle

Example 1: Solve the triangle: a = 3, b = 2, A = 400.

Example 2: Solve the triangle: a = 6, b = 8, A = 350.

Example 3: Solve the triangle: a = 2, c = 1, C = 500.

Draw a picture and then determine the number of triangles.

1. C=40°, b=19, c=20 2. B=67°, A=13, b=22 3. B=23°, a=9.2, C=65º

4. B=45°, a=28, b=20 5. C=27°, a=21, c=15 6. A=40°, a=10, c=18

7. A=36°, a=19, b=17 8. B=112°, a=12, b=11 9. A=27°, b=25, C=26°

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adj opp

θ

Law of Sines & Ambiguous Case (cont…)

Determine the number of Triangles. Do NOT Solve yet.

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Given Angle Goes HERE

Given Side opposite Given Angle Goes HERE

Practice - Law of Sines

A. Find the indicated measurement. (angles to the nearest tenth)

1. Find b if A=118°, B=22°, c=24 2. Find C if A=88°, a=26, b=16.1, c=21

3. Find c if B=44°, C=53°, b=7 4. Find C if A=82°, a=20, b=24, c=20

5. Find a if A=39°, C=51°, b=27 6. Find C if A=103°, a=26, b=24, c=6

7. Find c if B=101°, C=63°, a=9 8. Find A if B=129°, a=11, b=33, c=25

9. Find a if A=93°, C=58°, b=16 10. Find C if A=97°, a=22, b=9, c=19

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Determine the number of triangles. Show work for support. DO NOT SOLVE.

11. a=13.7, A=25°, B=78° 12. b=50, a=33, A=132°

13. A=38°, B=63°, c=15 14. a=125, A=25°, b=150

15. b=15.2, A=12°30’, C=57° 16. a=32, c=20, A=112°

17. b=795.1, c=775.6, B=51° 18. b=15, c=13, C=50°

19. a=12, b=15, A=55° 20. b=41, A=33°, B=29°

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Day 2 Homework Solve the following triangles. Round angles and lengths to nearest tenths.

1. 1. _______ Triangle(s) 2. _______ Triangle(s)

Triangle 1 Triangle 2

A, a

B, b

C, c

Triangle 1 Triangle 2

A, a

B, b

C, c

3. _______ Triangle(s)

Triangle 1 Triangle 2

A, a

B, b

C, c

4. _______ Triangle(s)

Triangle 1 Triangle 2

A, a

B, b

C, c

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Day 3 - Law of Cosines

*Used when not a right triangle and you have SSS or SAS from geometry

Example 1: Find all missing sides and angles.

Example 2: Find all missing sides and angles given B = 125˚, a = 19, and c = 15.

Practice: Solve the triangle given B = 62˚, a = 7, and c = 4.

Example 3: Solve triangle ABC given a = 8, b = 3, and c = 6.

Example 4: Solve the triangle given a = 13, b = 21, and c = 19.

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a2=b2+c2−(2bc )cos Ab2=a2+c2−(2ac )cosBc2=a2+b2−(2ab )cosC

Day 3 – Practice and HomeworkSolve each triangle. Round all answers to the nearest tenth.1. a=1.5, b=2.3, c=1.9 2. a=15, b=18, c=20

3. A=51°, b=40, c=45 4. a=20, b=12, c=28

5. A=52°, b=120, c=160 6. a=12.5, b=15.1, c=10.3

7. a=15, b=18, c=17 8. A=53°, b=40, c=49

9. A=42°, b=120, c=120 10. B=100°, a=10, c=8

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Lesson 4 -Area of a Triangle Notes

Area of a triangle when information is SAS: Area of a ∆ = 12absinC∨¿

12acsinB∨1

2bcsinA

Find the area of the following triangles, round final answer to the nearest tenth of a square unit.1. C = 49⁰, a = 16 cm, b = 25 cm 2. A = 47⁰, B = 65⁰, c = 19 cm

Heron’s Formula: use when given all 3 sides: Area of ∆ = √s (s−a ) ( s−b ) (s−c ) where s = semi-perimeter

s=12

(a+b+c )

Find the area of the triangle (final answer nearest tenth).3. a = 6 cm, b = 9 cm, c = 13 cm 4. a = 30 m, b = 50 m, c = 56 m

Find the area. Final answer nearest tenth.5. A = 18⁰, B = 69⁰, a = 14.2 6. c = 20 m, A = 45⁰ , B = 30⁰

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7. In Triangle ABC, AB = 14 meters, AC = 20 meters, and the area of the triangle is 82 square meters. Find the measure of angle A, rounded to the tenths.

8. The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of such land is $3 per square foot, how much does the land cost?

9. Farmer Bean has a triangular field with sides of 240 feet, 300 feet, and 360 feet. Through experimentation based off of past practice with similar compounds, he finds that Mrs. Smith uses great fertilizer! He wants to apply fertilizer to the field. If one 40-pound bag of Mrs. Smith’s fertilizer can fertilize 6,000 square feet, how many bags must he buy to cover the field?

10. To approximate the area of a lake, a surveyor walks around the perimeter of the lake, taking the measurements shown in the picture. Find the approximate area of the lake.

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Area of a Triangle Practice Find the area of each triangle to the nearest tenth. Show all work.

1. A=37°, B=69°, c=3.58 2. a=5, b=12, c=13

3. a=11, b=13, c=16 4. C=85°, a=2, B=19°

5. A=50°, b=12, c=14 6. B=25°, C=110°, b=14

7. A=115°, b=15, c=20 8. C=100°, a=68, c=110

9. a = 10, b = 12.4, c = 6 10. B = 11, c = 13, A = 74˚

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AFM Word ProblemsApplications of Right Triangle Trig (SOHCAHTOA), Law of Sines, and Law of CosinesMake a sketch with a triangle (not all are right triangles!), set up an equation and solve.

*1. A tree casts a shadow 21 meters long. The angle of elevation of the sun is 51o. What is the height of the tree?

2. From a point on the ground, the angle of elevation to the top of a tree is 34o. From a point 25 feet closer to the tree, the angle of elevation to the top of the tree is 48o. Find the height of the tree.

*3. You are flying a kite and have let out 80 meters of string. The kite’s angle of elevation with the ground is 40o. If the string is stretched straight, how high is the kite above the ground?

4. A 15 meter pole is leaning against a wall. The foot of the pole is 10m from the wall. Find the angle the pole makes with the ground.

*5. A ranger’s tower is located 44 m. from a tall tree. From the top of the tower, the angle of elevation to the top of the tree is 28 degrees and the angle of depression to the base of the tree is 36 degrees. How tall is the tree?

6. From the top of the lighthouse 210 feet high, the angle of depression of a boat is 27o. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level.

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7. A lighthouse operator at a point 80 m above sea level sights two sailboats. The angle of depression to one of the sailboats is 10o. The angle of depression to the second sailboat is 15o. How far apart are the two sailboats?

8. From a point A, the angle of elevation to the top of a building is 50°. From point B which is 11 m closer to the building the angle of elevation to the top is 63°. How far is point B from the top of the building? How tall is the building?

9. A field is triangular shaped with sides of lengths 83 m, 120 m, and 165 m. What is the measure of the angle formed by the two longest sides? What is the area of the field?

10. From a stationary hot air balloon 500 feet above the ground, two sightings of a lake are made. How long is the lake?

11. Anna and Elsa are standing 425 yards apart. They both see a helicopter in between them. Anna looks up at the helicopter at an angle of elevation of 43o and Elsa spots the helicopter at an angle of elevation of 38o. How high is the helicopter?

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