A) Use of Calculator

Alg 3 Ch 7.2, 8 11

7.2 Right Triangle Trig

A) Use of calculator

sin =

sin x = .4741

B) Create right triangles

1) If A = and b = 25, find a

2) If A = , a = 1

find b and c

C) Angle of elevation is the angle between the horizontal and the line of sight to the top of an object.

Angle of depression is determined by the horizontal line down to the line of sight.

1) How tall is a tree whose shadow is 47’ long when the angle of

elevation of the sun is .

2) One of the two congruent sides of an isosceles triangle is 23 cm and the vertex

angle is 43. How long is the base?

3) A balloon is floating between 2 people 50’ apart. The angle of elevation of the balloon from one is 63.5and the angle of elevation from the other is 32.6. How high is the balloon?

Ch 8.1 Law of Cosines

What happens if you don’t have a right triangle and you want to find sides and angles?

Law of Cosines: Given info on SSS or SAS you can find the remaining sides and angles.

Area of a Triangle Let K = Area y

K = by sinA =

K = bcSinA

K = acSin B

K = abSinC

EX. Find the area of the triangle,

Heron’s Formula: s = (a + b + c ) (no angles!)

Find the area of the triangle:

Law of Sines ASA, AAS

sin A = sin C =

c (sin A) = a (sin C

ex. Find a if A = 30, B = 45, and b = 8

ex. Find b if B = 60, c = 75, and a = 15

ex. To the nearest meter, find the distance across the pond if

8.2 Law of Sines SSA Three cases

Think of a wrecking ball where one side and one angle are set.

Case 1: no solution (not possible) (error message)

a < h

Case 2: one solution, two ways

Case 3: two solutions

Ex. Solve ABC if a = 50, b = 65 and A = 57 (SSA)

65(sin57) = 50(sinB)

54.5 = 50(sinB)

1.09 = sin B

(not possible, -1 sin B 1 )

Ex. Solve ABC if a = 60, b = 65 and A = 57 (SSA)

Ex. Solve ABC if c = 8, b = 20 and B = 122

Summary

Solve each of the following triangles. Express all answers correct to two decimal places.

(1) a = 5.7, b = 6.9 , g = 90°, ,,

(2) a = 13.32 , a = 18.7° , g = 90°

(3) a = 2.6 , b = 3.1 , c = 4.3

(4) a = 14 , c = 8.1 , b = 58.2°

(5) a = 28.3° , b = 61.3° , c = 40

(6) a = 51° , c = 18 , a = 25

Answers

(1) c = 8.95 , a = 39.56° , b = 50.44°

(2) b = 71.30° , b = 39.35 , c = 41.55

(3) a = 36.82° , b = 45.62° , g = 97.56°

(4) a = 86.52° , g = 35.28° , b = 11.92

(5) g = 90.40° , a = 18.96 , b = 35.09

(6) b = 94.98° , g = 34.02° , b = 32.05

1. Solve the following right triangles (ÐC is the right angle).

a) a = 2, b = 7 b) mÐA = 16º, c = 14

c) mÐB = 64º, c = 19.2 d) a = 9, mÐB = 49º

e) mÐB = 30º, b = 11

2. Solve the following triangles completely.

a) mÐA = 49º, mÐB = 57º, a = 8 b) mÐA = 83º, a = 80, b = 70

c) mÐB = 70º, mÐC = 58º, a = 84 d) a = 5, b =6, c = 7

e) mÐB = 47º, a = 20, b = 24 f) mÐA = 95º, a = 6, b = 8

g) mÐA = 58º, a = 26, b = 29

3. A shadow 30 m long is thrown from a tree. If the angle of depression of the sun is 65º, how tall is the tree?

4. The angle of elevation from a point on the ground to the top of a building is 38º. From a point 50 ft closer, the angle of elevation is 45º. How tall is the building?

5. A tree is broken by the wind. The top touches the ground 13 m from the base of the tree. It (the broken branch) makes an angle with the ground of 29º. How tall was the tree?

6. A triangular lot faces 2 streets that meet at an angle of 85º. The sides of the lot facing the streets are each 160 ft. Find the perimeter of the lot.

7. Two planes leave an airport at the same time, each flying 110 mi/hr. One flies 60º east of north, the other flies 40º east of south. How far apart are the planes after 3 hours?

8. The sides of a triangle are 6.8 cm, 8.4 cm and 4.9 cm. Find the measure of the smallest angle.

9. A 40 ft antenna stands on top of a building. From the ground, the angles of elevation of the top and bottom of the antenna are 56º and 42º respectively. How tall is the building?

Part 2

1. A balloon is floating between two observers who are 220 feet apart. The angle of elevation of the balloon from observer one is 67º and from observer two is 31º. How far is the balloon from observer one?

2. To find the distance across a canyon, a surveying team locates points A and B on one side of the canyon and point C on the other side of the canyon. The distanced between A and B is 85 yards. mÐCAB is 68º and mÐCBA is 88º. Find the distance across the canyon.

3. The longer side of a parallelogram is 6.00

meters. mÐA = 56º and mÐα= 35º. Find

the length of the longer diagonal.

4. Two observers in line directly under a kite, and 30 ft. apart, observe the kite at angle of elevations of 62º and 78º respectively. Find how high the kite is in the air.

5. A 35 foot high telephone pole is situated on an 11º slope from A. The

angle of elevation from point A to the

top of the pole is 32º. Find the length

of the wire AC.

6. A surveying team determines the height of a hill by placing a 23 ft pole at the top and measuring the angles of elevation to the bottom and the top of the pole. If they are 70º and 75º respectively, find the height of the hill.

7. A developer has a triangular lot at the intersection of two streets. The streets meet at an angle of 72º and the lot has 300 ft of frontage along one street and 416 ft of frontage along the other street. Find the length of the third side of the lot.

8. The sides of a triangular city lot have sides of 224 ft, 182 ft and 165 ft. Find the smallest angle.

9. Find the number of acres in a pasture whose shape is a triangle with sides 800 ft, 1020 ft and 680 ft. (Hmm…how many square feet in an acre?)

Answers:

a) c=7.28, mÐA=15.9, mÐB=74.1 b) mÐB = 74, a = 3.86, b = 13.5

c) mÐA = 26, a = 8.42, b = 17.3 d) b = 10.3, c = 13.7 , mÐA = 41º

e) mÐB = 60, a = 19.1, c = 2.2

2. Solve the following triangles completely.

a) mÐC = 74º, b = 8.89, c = 10.2, A = 34,2

b) mÐB = 60.3º, mÐC = 36.7, c = 48.2, A = 1670

c) mÐA = 52º, b = 100, c = 90.4, A 3570

d) mÐA = 44.4, mÐ57.1, mÐC = 78.5, A = 14.7

e) mÐA = 37.6º, mÐC = 95.4, c = 32.7, A = 239

f) no triangle

g) mÐB = 71.1, mÐC = 50.9, c = 23.8, A = 293

mÐB’ = 108.9, mÐC’ = 13.1, c’ = 6.95, A’ = 85.4

3. h = 64.3 4. h = 179 ft

5. h = 22.1 m 6. p = 536 ft

7. d = 424 mi 8. 35.7˚

9. h = 61.9 ft

Part 2

1. 114 ft 2. 194 yd

3. 8.67 m 4. 40.3 ft

5. 95.9 ft. 6. 64.2 ft

7. 431 ft 8. 46.5˚

9. 271558 square ft = 6.23 acres

3. The longer side of a parallelogram is 6.00

meters. mÐA = 56º and mÐα= 35º. Find

the length of the longer diagonal.

4. Two observers in line directly under a kite, and 30 ft. apart, observe the kite at angle of elevations of 62º and 78º respectively. Find how high the kite is in the air.

8. The sides of a triangular city lot have sides of 224 ft, 182 ft and 165 ft. Find the smallest angle.

9. Find the number of square feet in a pasture whose shape is a triangle with sides 800 ft, 1020 ft and 680 ft.

Answers:

1. 114

2. 194

3. 8.67

4. 40.3

6. 64.2

7. 431

8. 46.5

9 Area= 271558