Download Geometry 2 – Unit Three: Trigonometry, Practice

Geometry 2 – Unit Three: Trigonometry, Practice
In Problems #1 - #6, express each trigonometric ratio as a fraction.
1.
sin P = __________
P
2.
cos P = __________
7
3.
tan P = __________
Q
4.
sin R = __________
5.
cos R = __________
6.
25
R
24
tan R = __________
In Problems #7 - #10, solve each right triangle. Round answers to the nearest tenth.
AC = __________
7.
8.
∠ D = __________°
∠ A = __________°
DF = __________
∠ C = __________°
FE = __________
A
D
7
9.5
25°
B
C
10.5
HK = __________
9.
F
10.
E
∠ L = __________°
KG = __________
∠ N = __________°
∠ G = __________°
MN = __________
G
L
22
6.7
8.3
39°
K
H
M
N
11.
A man leans a ladder up against a building. The ladder forms a 75° angle with the ground. The
ladder reaches to a point 19.3 feet above the ground. How long is the ladder? Round the answer
to the nearest whole number.
12.
A woman leans a 16 foot ladder against the side of a building. She positions the ladder so that
the bottom of the ladder is 2 feet away from the building. What is the measure of the angle
formed by the ladder and the ground? Round the answer to the nearest whole number.
13.
A man leans a 12 foot ladder against the side of a building. The ladder forms an 78° angle with
the ground. How far up the side of the building does the ladder reach? Round the answer to the
nearest tenth.
14.
A forest ranger in a 90 foot observation tower sees a fire. The forest ranger must look down at
an angle of 7° to see the fire. What is the horizontal distance between the base of the tower and
the fire? Round the answer to the nearest whole number.
15.
A forest ranger in an observation tower sees a fire. The forest ranger must look down at an
angle of 10° to see the fire. The horizontal distance between the base of the tower and the fire
is 850 feet. How tall is the observation tower? Round the answer to the nearest whole number.
16.
When the angle of elevation to the sun is 37°, a flagpole casts a shadow that is 24.2 feet long.
What is the height of the flagpole? Round the answer to the nearest tenth.
17.
When the angle of elevation to the sun is 28°, a 30 foot flagpole casts a shadow. What is the
length of the shadow? Round the answer to the nearest tenth.
18.
A 45 foot flagpole casts a 15.8 foot shadow. What is the measure of angle of elevation to the
sun? Round the answer to the nearest tenth.
19.
An air traffic controller at an airport sights a plane at an angle of elevation of 12°. The pilot
reports that the plane’s altitude is 4,500 feet. What is the horizontal distance between the point
directly beneath the plane to the airport? Round the answer to the nearest whole number.
20.
A plane sights the end of a runway by looking down at an angle of depression of 5°. The
distance from the point directly beneath the plane to the end of the runway is 12,500 feet. What
is the altitude of the plane? Round the answer to the nearest whole number.
In Problems #21 - #22, find the approximate area of the triangle. Use the formula
Area = ½ * b * h. Round your answer to the nearest tenth.
21.
22.
77°
26°
**************************ANSWERS**************************
opposite leg
24
→ QR is opposite ∠ P, PR is the hypotenuse → sin P =
hypotenuse
25
1)
sine =
2)
cosine =
adjacent leg
7
→ PQ is adjacent to ∠ P, PR is the hypotenuse → cos P =
hypotenuse
25
3)
tangent =
opposite leg
24
→ QR is opposite ∠ P, PQ is adjacent to ∠ P → tan P =
adjacent leg
7
4)
sine =
5)
cosine =
adjacent leg
24
→ QR is adjacent to ∠ R, PR is the hypotenuse → cos R =
hypotenuse
25
6)
tangent =
opposite leg
7
→ PQ is opposite ∠ R, QR is adjacent to ∠ R → tan R =
adjacent leg
24
7)
To find AC , use the Pythagorean Theorem → 7 2 + 10.52 = x 2 → 49 + 110.25 = x 2 →
opposite leg
7
→ PQ is opposite ∠ R, PR is the hypotenuse → sin R =
hypotenuse
25
159.25 = x 2 → 159.25 = x 2 → 12.619 ≈ x → AC ≈ 12.6
10.5
→ tan A = 1.5 → tan −1 (1.5 ) ≈ 56.3099 → ∠ A ≈ 56.3°
To find ∠ A, use tan (opp/adj) → tan A =
7
To find ∠ C , use tan (opp/adj) → tan C =
8)
7
→ tan C = 0.6667 → tan −1 ( 0.6667 ) ≈ 33.6901 → ∠ C ≈ 33.7°
10.5
∠ D = 180 − 90 − 25 = 65°
x
x
→ 0.4226 =
→ 0.4226 ⋅ 9.5 = x → 4.0147 ≈ x → DF ≈ 4.0
9.5
9.5
y
y
To find FE , use cos (adj/hyp) → cos 25° =
→ 0.9063 =
→ 0.9063 ⋅ 9.5 = y → 8.60985 ≈ y → FE ≈ 8.6
9.5
9.5
To find DF , use sin (opp/hyp) → sin 25° =
9)
∠ G = 180 − 90 − 39 = 51°
22
→ 0.8098 =
x
22
To find KG, use sin (opp/hyp) → sin 39° =
→ 0.6293 =
y
To find KH , use tan (opp/adj) → tan 39° =
10)
22
22
→x=
→ x ≈ 27.168 → HK ≈ 27.2
x
0.8098
22
22
→y=
→ y ≈ 34.958 → KG ≈ 35.0
y
0.6293
To find MN , use the Pythagorean Theorem → x 2 + 6.7 2 = 8.32 → x 2 + 44.89 = 68.89 →
x 2 = 24 → x 2 = 24 → x ≈ 4.8990 → MN ≈ 4.9
6.7
→ cos L = 0.8072 → cos −1 ( 0.8072 ) ≈ 36.1739 → ∠ L ≈ 36.2°
8.3
6.7
→ sin N = 0.8072 → sin −1 ( 0.8072 ) ≈ 53.8261 → ∠ N ≈ 53.8°
To find ∠ N , use sin (opp/hyp) → sin N =
8.3
To find ∠ L, use cos (adj/hyp) → cos L =
11)
19.3
19.3
→ 0.9659 =
→
x
x
19.3
x=
→ x ≈ 19.98 → Ladder ≈ 20 feet
0.9659
sin 75° =
x
12)
cos x° =
2
→ cos x° = 0.125 →
16
cos −1 ( 0.125 ) = 82.8192 → x ≈ 82.82 →
Angle ≈ 83°
16 feet
19.3 feet
x°
75°
13)
2 feet
x
x
→ 0.9781 = →
12
12
12 ⋅ 0.9781 = x → 11.738 ≈ x → Height reached ≈ 11.7 feet
sin 78° =
12 feet x
78°
14)
If the angle of depression
is 7°, then the angle of
elevation is also 7°.
7°
90 feet
7°
x
90
90
90
tan 7° =
→ 0.1228 =
→ x=
→ x ≈ 732.991 → Horizontal distance ≈ 733 feet
x
x
0.1228
15)
If the angle of depression
is 10°, then the angle of
elevation is also 10°.
10°
x
10°
850 feet
x
x
tan10° =
→ 0.1763 =
→ 0.1763 ⋅ 850 = x → 149.878 ≈ x → Tower ≈ 150 feet
850
850
16)
tan 37° =
x
x
→ 0.7536 =
→
24.2
24.2
17)
tan 28° =
30
30
→ 0.5317 =
→
x
x
30
→ x ≈ 56.422 →
0.5317
Shadow ≈ 56.4 feet
0.7536 ⋅ 24.2 = x → 18.236 ≈ x →
x=
Flagpole ≈ 18.2 feet
SUN
SUN
30 feet
x
28°
x
37°
24.2 foot shadow
18)
45
→ tan x° = 2.8481 →
15.8
tan −1 ( 2.8481) ≈ 70.653 →
tan x° =
19)
If the angle of depression is 12°, then
the angle of elevation is also 12°
4500
4500
→ 0.2126 =
→
x
x
4500
→ x ≈ 21,170.835 →
x=
0.2126
Horizontal Distance ≈ 21,171 feet
Angle of elevation ≈ 70.7°
tan12° =
SUN
12°
PLANE
4500 feet
12°
45 feet
x
x°
15.8 feet
20)
If the angle of depression is 5°, then the angle of elevation is also 5° → tan 5° =
0.08749 =
x
→
12,500
x
→ 0.08749 ⋅ 12,500 = x → 1,093.608 ≈ x → Altitude ≈ 1,094 feet
12,500
5°
PLANE
x
5°
12,500 feet
21)
22)
x
x
→ 0.4877 = → 0.4877 ⋅ 9 = x → 4.3896 ≈ x →
9
9
1
1
To find the area, use the formula A = * b * h → A = * 9 * 4.3896 → A ≈ 19.7532 → A ≈ 19.8 cm 2
2
2
To find x, use tan (opp/adj) → tan 26° =
15
15
15
→ 4.3318 = → x =
→ x ≈ 3.4630
x
x
4.3318
1
1
To find the area, use the formula A = * b * h → A = *15*3.4630 → A ≈ 25.9725 → A ≈ 26.0 in 2
2
2
To find x, use tan (opp/adj) → tan 77° =