Trigonometry (Height And Distance)

Line of sight

When the human eye see any object, then the straight line between human eye and the object is called the line of sight.

Angle of elevation

When the human eye see an object above the horizontal level then the line 0f sight makes an angle with the horizontal level . This angle is called angle of elevation.

In the given figure \theta is the angle of elevation.

Angle of depression

When the human eye see an object below the horizontal level then the line 0f sight makes an angle with the horizontal level . This angle is called angle of depression.

In the given figure \theta is the angle of depression.

 

Question : There is a point, 60 m away from the foot of the tower and the angle of elevation from the top of a  tower to that point is 60 degree. Find the height of the tower

Solution In the figure, AB is a tower and C is a point on the ground

In \triangleABC

\frac{AB}{BC} = tan 60

\frac{AB}{60}\sqrt{3}

AB = 60\sqrt{3}

height of tower = 60\sqrt{3}

 

Question A tree broke in the storm and broken part bends in a way such that the top of the tree touch the ground at a point which is 10 m away from the foot of the tree and makes an angle of 45 degree with the ground level. Find the height of the tree

Solution

In \triangleABC

\frac{AB}{BC} = tan 45

\frac{AB}{8} = 1

AB = 8 m

And

\frac{BC}{AC} = cos 45

\frac{8}{AC} = \frac{1}{\sqrt{2}}

AC = 8\sqrt{2} m

Height of tree = AB + BC = 8 +  8\sqrt{2} m = 8(\sqrt{2} + 1) m

 

Question The shadow of a tower is \frac{1}{\sqrt{3}} times its height. Find the angle of elevation from the top of the tower to the end point of the shadow

Solution

Let the height of the tower be x , then the length of shadow = \frac{1}{\sqrt{3}}x

In \triangleABC

\frac{AB}{BC} = tan \theta \frac{x}{\sqrt{3}x} = tan \theta

tan \theta\frac{1}{\sqrt{3}} \theta = 60 degree

 

Question The angle of elevation of an aeroplane from a point on the ground is 60 degree. After 15 seconds , the elevation changes to 30 degree. If the aeroplane is flying at a height of 1500\sqrt{3} m, find the speed of the plane.

Solution

In \triangleABC

\frac{AB}{BC} = tan 30

\frac{1500\sqrt{3}}{BC} = \frac{1}{\sqrt{3}}

BC = 4500 m

In \triangleCDE

\frac{DE}{CE} = tan 60

\frac{1500\sqrt{3}}{CE} = {\sqrt{3}}

CE = 1500 m

BE = AC – CE = 4500 – 1500 = 3000 m

In 15 seconds, plane cover a distance of 3000 m.

speed of plane = 200 m / sec.

 

                                  Problems for practise 

1. The shadow of a 90 feet high poll is 72 feet long and at the same time, the shadow of a man is 5.6 feet long. Find the height of a man ?

a) 6 feet                        b) 7 feet                       c) 6.5 feet                          d) 7.5 feet

Answer – b) 7 feet

 

2.  A ladder is resting against the wall at a height of 10 m. and the angle between the ladder and ground is 60 degree. if the ladder slips to an angle of 30 degree, find the height of the wall at which the ladder is resting now ?

a) 20 /3 m                       b) 10 / 3 m                           c) 16 /3 m                                     d) 5 m

Answer – a) 20 / 3 m

 

3. A tower is fixed at the top of a building. there is a point on the ground which is 30 m away from the foot of the building and the angle of elevation from this point to the top and bottom of the tower are 45 degree  and 60 degree respectively. find the height of the tower

a) 17.64                           b) 18.56                         c) 20.48                        d) 21.96

Answer – d) 21.96

 

4. The angle of depression from the top of a building to a point on the ground is 30 degree. If the height of the building is 50 meter , then find the approximate distance from that point to the foot of the building.

a) 87 meter                b) 82 meter                   c) 91 meter                      d) 76 meter

Answer – b) 82 meter

 

5. A boy is flying a kite with tight string of length 250 meter. If the angle of elevation of the kite is 60 degree, then the vertical height of the kite is

a) 200 meter                   b) 240. 5 meter                          c) 225.5 meter                          d) 216.5 meter

Answer – d) 216.5 meter

 

6. The shadow of an electric pole is formed to be 20 meter longer, when the sun’s altitude is 30 degree than when it is 45 degree. Find the height of the pole.

a) 27.3 meter                    b) 22.5 meter                     c) 31.2 meter                             d) 16.8 meter

Answer – a) 27.3 meter

 

7. The angle of elevation from the top of a tower to two points on the ground which are a meter and b meter away respectively from the foot of the tower are complementary. Find the height of the tower

a) ab meter                 b) a/b meter                          c)\sqrt{ab}                       d) a^{2}b^{2}

Answer – c)\sqrt{ab}

 

8.There are two trees on the both sides of river of height 20 meter and 30 meter respectively and the angle of elevation from the top of the trees to a point on the surface of river are 30 and 60 degree respectively. Find the approximate width of the river

a) 59 meter                    b) 52 meter                   c) 48 meter                        d) 42 meter

Answer – b) 52 meter

 

9. The angle of depression from the top of a building of height 40\sqrt{3} meter to the top of tower of height 10\sqrt{3} meter is 30 degree. find the distance between the foot of the tower and the foot of the building.

a) 30 meter                        b) 40 meter                           c) 60 meter                              d) 90 meter

Answer – d) 90 meter

 

10. From the top and bottom of a pillar of height 120 meter to the top of a hill, the angle of elevation are 45 degree and 60 degree respectively. the height of the hill is

a) 164 meter                    b) 180 meter                        c) 240 meter                        d) 300 meter

Answer – a) 164 meter

 

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