1. A 15 foot-ladder makes a 52° angle w/ the ground. How far will the tip of the ladder be above the ground?

Given:

15 ft - ladder

52o – angle of ladder w/ the ground

sin Θ= opp/hyp

sin 52o=opp/15 ft

opp=sin 52o (15ft)

opp= 11.82 ft

2. Estimate the angle of elevation of the sun if a boy 5 feet tall cast a shadow 4 feet long on the level ground.

Given:

5 ft – height of the boy

4 ft – shadow cast by the boy

tan Θ=opp/hyp

tan Θ=5 ft/4 ft

tanΘ=1.25 or 51o 50’

3. A man wire on a telephone pole makes an angle of 64° with the ground. The end of the wire is 12 meters above the ground. How long is the wire?

Given:

12m – end of wire above the ground

64o – angle of wire with the ground

sin Θ=opp/hyp

sin 64o=12m/hyp

hyp=12m/sin 64o

hyp=13.35m

4. An airplane is flying at an altitude of 10,000ft. The angle of depression from the plane to a tree is 13o40’. How far horizontally must the plane fly to be directly over the tree?

Given:

10,000 ft – altitude of the plane

13o40’ – angle of depression from the plane to a tree

tan Θ=opp/adj

tan 13o40’=opp/10000 ft

opp=tan 13o40’ (10000 ft)

opp= 2431.57 ft

5. The angle of depression from the top of the building to a point on the ground is 34°50’ from the top if the building is 368m high?

Given:

368m – height of the building

34o50’ – angle of depression from the top of the building to a point on the ground

cos Θ=adj/hyp

cos 34o50’=368m/hyp

hyp=368m/cos 34o50’

hyp=448.34m

6. Suppose the angle of elevation of the sun is 28.4°. Find the length of the shadow cast by a man 6 ft. tall.

Given:

6 ft – shadow cast by a man

28.4o – angle of elevation of the sun

tan Θ=opp/adj

tan 28.4o=6 ft/adj

adj=6 ft/tan 28.4o

adj=11.10 ft

7. Find the angle of elevation of the sun if a 53.9 ft flagpole cast a shadow 74.6 ft long.

Given:

53.9 ft – height of the flagpole

74.9 ft – shadow cast by a flagpole

tan Θ=opp/adj

tan Θ=53.9 ft/74.6 ft

tan Θ=0.7225 or 35o50’

8. When Francis stands 127ft from the base of a bell tower, the angle of elevation to the top is 21°. Find the height of the building.

Given:

127 ft – distance from francis to the base of the bell tower

21o – angle of elevation

tan Θ=opp/adj

tan 21o=opp/127 ft

opp= tan 21o (127 ft)

opp=48.75 ft

9. From the lighthouse 75.3ft above the level of the ocean, the angle of depression of a boat is 23°40. How far is the boat from a point at water level directly under the point of observation?

Given:

75.3 ft – lighthouse above level of the ocean

23o40’ – angle of depression of the boat

cos Θ=adj/hyp

cos 23o40’=75.3 ft/hyp

hyp=75.3 ft/cos 23o40’

hyp=82.22 ft

10. The string on a kite is taut and makes an angle of 54°20 w/ the horizontal level ground if 85 meters of strings are out and the end of the string is held 1.5m above the ground.

Given:

85m – string of the kite

54o20’ – angle of string of kite w/ the ground

1.5m – boy held above the ground

sin Θ=opp/adj

sin 54o20’=(opp/85m) +1.5m

opp=(sin 54o20’(85m)) +1.5m

opp=69.06m + 1.5m

opp=70.56m

11. A ladder 16m long is placed so that foot is 3m from a building. How far must the foot of the ladder be moved from the building to lower the top of the ladder by 2m?

Given:

16m – ladder

3m – foot of the ladder from the building

2m – subtracted from the building to lower the top of the ladder

a2 + b2 = c2 a= 13.71m

a2 + 32 = 162 a2+ b2 = c2

a2 = 162 – 3 2 13.71 2 + b2 = 16 2

= 256 – 9 b2= 16 2 – 13.71 2

√a2 = √247 b2= 256 – 188

a= 15.71 √b2=√68

a= 15.71 – 2 b= 8.246 – 3

b= 5.246m the foot of ladder be moved from the building.

12. The angle of elevation from the top of an office building in Makati to the top of the RBA appliance center is 68°, while the angle of depression from the top of the office building to the bottom of 63°. The office building is 29ft from the RBA appliance center. Find the height if the RBA appliance center.

Given:

29 ft – distance of the office building from the appliance center

68o – angle of elevation of the office building to the top of the appliance center

63o – angle of depression of the office building to the bottom of appliance center

Angle of elevation Angle of depression

tan Θ= opp/adj tan Θ=opp/adj

tan 68o= opp/29 ft tan 63o=opp/29 ft

opp=tan 68o (29 ft) opp=tan 63o (29 ft)

opp=71.78 opp=56.92

height of the building = x1 + x2

H= 71.78 + 56.92 H= 128.7

13. The angle of elevation of the sun is 35°10’ and the shadow of the flagpole on horizontal ground is 20m. How tall is the flagpole?

Given:

20m – shadow cast by a flagpole

35o10’ – angle of elevation of the sun

tan Θ=opp/adj

tan Θ=opp/adj

tan 35o10’=opp/20m

opp=tan 35o10’ (20m)

opp=14.09m

14. An observer on top of a 25.6m building finds that the angle of depression of an open manhole is 42°. How far is the manhole from the base of the building?

Given:

25.6m – height of the building

42o – angle of depression of an open manhole

tan Θ=opp/adj

tan 42o=opp/25.6m

opp=tan 42o (25.6m)

opp=23.05m

15. A farmer has a right triangle shaped lot. The longest side is 42.3m and makes an angle of 52°20’ with the shortest side. What is the perimeter of the lot?

Given:

42.3m – longest side of a right triangle lot

52o20’ – angle made by the longest side and shortest side

cos Θ=adj/hyp P=A+B+C

cos 52o20’=adj/42.3m P=33.48 + 25.85 +42.3

adj=cos 52o20’ (42.3m) P=101.6m

adj=25.85m

a2 + b2 = c2

a2 + 25.852 = 42.32

a2 = 42.32 – 25.852

a2 = 1789.29 – 688.22

√a2 = √1121.06

a = 33.48

16. Find the angle of elevation of a 98.6m building at a point 35m from the foot of the building.

Given:

98.6m – height of the building

35m – distance of the point to the foot of the building

tan Θ=opp/adj

tan Θ=98.6m/35m

tan Θ=2.8171 or70o30’

17. What is the angle of elevation of the sun when a radio tower 206m high cast a shadow of 85m long?

Given:

206m – height of the radio tower

85m – shadow cast by the radio tower

tan Θ=opp/adj

tan Θ=206m/85m

tan Θ=2.4235 or 67o40’

18. The string of a kite is tied on a peg and makes an angle of 68°30’ w/ the ground. If a point directly below the kite is 216m from the peg, How high is the kite and how long is the string?

Given:

216m – distance below the kite from the peg

68o30’ – angle made of string of kite on a peg

Height of the kite Length of the string

tan Θ=opp/adj cos Θ=adj/hyp

tan 68o30’=opp/216m cos 68o30’=216m/hyp

opp=tan 68o30’ (216m) hyp=216m/cos 68o30’

opp=548.34m hyp=589.35m

19. A man on the light house 75m tall above sea level observed that the angle if depression of a boat is 38°20. How far is the boat from the foot to the lighthouse?

Given:

75m – height the lighthouse

38o20’ – angle of depression of the boat

tan Θ=opp/adj

tan 38o20’=opp/75m

opp=tan 38o20’ (75m)

opp=59.30m

20. Two hunters A and B sight the same prey on top of a hill is 300m high and the two hunters are on the same ground level, What is the shortest distance between the two hunters? What is the largest distance between them?

21. The angle of elevation of the top of the flagpole from a point on the same horizontal plane as the base of the flagpole was found to be 47°, and the angle of elevation of the flag is 33° from the same point when the flag is 18m up the flagpole. How tall is the flagpole?

Given:

18m – distance of flag up the flagpole

47o – angle of elevation from a point on the horizontal plane to the top of flagpole

33o – angle of elevation of the flag

tan Θ= opp/adj

tan 43o=opp/x+18 tan 57o=opp/x

opp= tan 43o (x +18) opp=tan 57o (x)

tan 43o (x + 18) = tan 57o (x)

x tan 43o + 18 tan 43o = x tan 57o

x tan 43o – x tan 57 = - 18 tan 43o

x = -18 tan 43o/tan 43o – tan 57o

x = -16.78/-0.6073

x = 27.63m

height of the flagpole= x + 18

H = 27.63m + 18m H = 45.63m

22. Each equal sides of an isosceles trapezoid is 48.6m and the parallel sides are 26.5m and 43.8m. What is the measure of each angle of the trapezoid? What is its altitude?

Given:

48.6m – two equal side of trapezoid

26.5m – smaller parallel side of trapezoid

43.8m – bigger parallel side of trapezoid

to find the adjacent side: the altitude of trapezoid:

(43.8m – 26.5m)/2 a2 + b2 = c2

= 8.65m 8.652 + b2 = 48.62

b2 = 48.62 – 8.652

cos Θ=adj/hyp b2 = 2361.96 – 74.82

cos Θ=8.65/48.6 √b2 = √2287.14

cos Θ=0.1779 or 79o50’ b = 47.82m

to find the other angle:

x = 180o – 79.83o

x = 100.16o

23. If the sun is 25° above the horizon. Find the length of a shadow cast by a building that is 500ft tall.

Given:

500 ft – height of the building

25o – angle made by the sun

tan Θ=opp/adj

tan 25o=500 ft/adj

adj=500 ft/tan 25o

adj=1072.25m

24.

a. the bottom B of a 16ft ladder is placed 3ft from a wall as shown at the right. How far is the top of the ladder

from the ground C?

Given:

16 ft - ladder

3 ft – distance of wall to the foot of the ladder

a2 + b2 = c2 √b2 = √247

32 + b2 = 162 b = 15.71ft

b2 = 162 – 32

b2 = 256 - 9

b. the bottom of the ladder is more 2 additional ft. away from the wall. How many feet do A slide down.

a2 + b2 = c2 b = 15.19ft

(3 + 2)2 + b2 = 162 b = 15.71 – 15.19

b2 = 162 – 52 b = 0.52ft A slide down

b2 = 256 – 25

√b2 = √231

25. A foot railroad track expand 6 inch because the day is very hot. This causes end L to break off and more to pollution forming a right triangle RLD. Find DL

26. Find the length of the longest diagonal BG of the next angular solid shown at the right.

27. Two tangent segments to a circle intersect and make an angle of 48°15’12”. Find the radius of the circle if one of the tangent segments is 27.2cm

28. The diameter of a circle is 15.04 cm, two tangents to the circle intersect and make an angle of 50°14’. How long are the tangent segments?

Given:

15.04 cm – Diameter of the circle

50o14’ – angle made by two tangent segment

To find the opposite side: sin Θ=opp/hyp

opp=15.04/2 sin 25.12o=7.52 cm/hyp

opp=7.52 hyp=7.52 cm/sin 25.12o

hyp=17.71 cm

to find the angle of reference:

Θ=50o14’/2

Θ=25.12o

29. An arc of a circle is 8.8cm.Find the central angle of the chord if its radius is 10.5cm.

Given:

8.8cm – arc length

10.5cm – radius of the circle

Θ=arc length (s)/radius (r)

Θ=8.8 cm/10.5 cm

Θ=0.8380

0.8380 (180/π)

central angle = 48.01o

30. A regular pentagon is inscribed in a circle w/ a radius of 15 cm. Find the perimeter of the pentagon.

Given:

15 cm – radius of the circle

31. The dimensions of a rectangle are 92cm and 164cm. Find the angle made by the diagonal and the shorter side.

32. When the angle of elevation of the sun is 61° the building cast a shadow of 50 meters. How high is the building?

Given:

61o – angle of elevation of the sun

50m – shadow cast by the building

tan Θ=opp/adj

tan 61o=opp/50m

opp=tan 61o (50m)

opp=90.20m

33. From the top of a tower w/c is 175ft tall, the angle of depression to a house is 13°. How far is the house from the tower?

Given:

175 ft – height of the tower

13o – angle of depression to a house

tan Θ=opp/adj

tan 13o=opp/175

opp=tan 13o (175)

opp=40.40 ft