**Heights and Distances** are the main application of trigonometry, which is extensively used in the real life. Trigonometry is useful to astronomers, navigators, architects, and surveyors, etc. in solving problems related to heights and distances.

Before going into the problems there are a few terminologies that we used to solve the problem. They are:

**Line of sight**: It is the line drawn from the eye of an observer to the point in the object viewed by the observer.**Angle of impression:**The angle between the horizontal and the line of sight joining an observation point to an elevated object is called the angle of elevation**.**

**Angle of depression:**The angle between the horizontal and the line of sight joining an observation point to an object below the horizontal level is called the angle of depression.

**Sample Problems on Height and Distance**

**Problem 1:** If a pole 6 m high casts a shadow 2√3 m long on the ground, find the Sun’s elevation.

**Solution:**

Let

ABbe the pole which is of height6 m.Let

BCbe the shadow of the building2√3.Now, in ∆ ABC,

tan θ = AB / BC

=> tan θ = 6 / 2√3

Now, simplifying using rationalization

=> tan θ = (3 / √3)*(√3 / √3)

=> tan θ = 1 / √3

=> θ = tan

^{-1}(1 / √3)Hence, θ = 60

^{o}Therefore, sun’s elevation from the ground is

60.^{o}

**Problem 2:** An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.

**Solution:**

Let

PQbe the height of the observer of1.5 m.Let

ABbe the height of the tower of22 m.And, let

QBbe the horizontal distance between the observer and the tower=> PQ = MB = 1.5 m

=> AM = AB – MB

=> AM = 22 – 1.5 = 20.5

Now, in

∆APM,=> tan θ = AM / PM

=> tan θ = 20.5 / 20.5

=> tan θ = 1

=> θ = tan

^{-1}(1 )Hence, θ = 45

^{o}Therefore, the angle of elevation of the top of the tower from the eye of the observer is

45^{o}

**Problem 3:** An airplane is flying h meters above the ground. At a particular instant, the angle of elevation of the plane from the eyes of a boy sitting on the ground is 60°. After some time, the angle of elevation changed to 30°. Find the distance covered by the plane during that time assuming it traveled in a straight line.

**Solution:**

Let

xbe the horizontal distance between the observer and plane at the first instant.Let

ybe the horizontal distance between the observer and plane at the second instant.And, BA = CD = h

In

∆OAB,=> tan 60° = AB / OA

=> √3 = h / x

=> x = h / √3

In ∆ OCD,

=> tan 30° = CD / OD

=> 1/√3 = h / (x+y)

=> x + y = √3h

Distance travelled by plane = AD = y

=> (x + y) − x = √3h − h / √3

=> y = (2 / √3)h

So, if the airplane is flying h meters above the ground, it would travel for (2/√3) h meters as the angle of elevation changes from

60°to30°.

**Problem 4:** From the top of the tower 30 m height a man is observing the base of a tree at an angle of depression measuring 30 degrees. Find the distance between the tree and the tower.

**Solution:**

In the above diagram

ABrepresents the height of the tower,BCrepresents the distance between the foot of the tower and the foot of the tree.Now we need to find the distance between the foot of the tower and the foot of the tree (BC). For that as angle of depression is given so by vertically opposite angle property of triangle

∠CAD = ∠BCAIn

∆BCA,=> tan θ = Opposite side / Adjacent side

=> tan 30° = AB / BC

=> 1/√3 = 30 / BC

=> BC = 30√3

=> BC = 30 (1.732) [Approximately]

=> BC = 51.96 m

So, the distance between the tree and the tower is

51.96 m.

**Problem 5:** From the top of a building 30 m high, the top and bottom of a tower are observed to have angles of depression 30° and 45° respectively. Find the height of the tower.

**Solution:**

Let

ABbe the building andCDbe the tower.The angle of depressions is given 30° and 45° to the top and bottom of the tower. So by vertically opposite triangle property

∠FBD = ∠EDB and ∠FBC = ∠ACB.Now, AB = 30 m. Let DC = x.

Draw DE perpendicular AB. Then AE = CD = x.

Therefore BE = (30 – x) m.

In

∆ACB,=> cot θ = Adjacent side / Opposite side

=> cot θ = AC / AB

=> cot 45° = AC / 30

=> AC = 30 [cot 45° = 1]

So, DE = AC = 30 m

In

∆EDB,=> tan θ = Opposite side / Adjacent side

=> tan 30° = BE / DE

=> 1/√3 = BE / 30

=> BE = 30 / √3

=> CD = AE = AB – BE = 30 – (30 / √3)

=> 30[1 – (1 / √3) ] m

Height of the tower is

30[1 – (1 / √3) ] m

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