LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

TRIGONOMETRY- CORE CONCEPTS OF HEIGHT & DISTANCE

**HEIGHT &
DISTANCE –**

**This chapter deals with very practical chapter of
Trigonometry in determining such parameters as height of objects like a
mountain, a tree, a tower, tall building, a tall post, or a pillar, etc., and
breadth of a river, etc., which are otherwise very difficult to measure.**

**Some
Important Definition –**

__Line of Sight –__
When the eye of a person at a point O looks at an object A, then the line OA is
called the line of sight.

**(i) Angle of Elevation – **

**Suppose a man from a point O, looks up at an object A,
placed above the level of his eye. Then, the angle which the line of sight
makes with the horizontal through O, is called the Angle of Elevation of A as seen from O. **

**Let, OB be a horizontal line. Let a man at O on the level
ground be looking up towards an object P, say an aeroplane or the top of a
tower or the flag at the top of a building.**

**Then, ∠BOA is the angle of elevation of P as seen from O.**

**(ii) Angle of Depression –**

**Suppose a man from a point O, looks down at an object
A, placed below the level of his eye. Then, the angle which the line of sight
makes with the horizontal through O, is called the angle of depression of A as seen from O.**

**Let, BO be a horizontal line. Let a man at O, on
the top of a tower be looking down
towards an object A, say a ship in the sea.**

**Then, ∠BOA is the angle of depression of A as seen from O.**

__An Important Information –__ Angle of depression of
A as seen from O = Angle of elevation of O as seen from A.

**So, ∠BOA = ∠CAO**

**There are some examples are given below for your
better understanding -**

**Example.1) The angle of elevation of
the top of a tower at a distance of 90 meters from its foot on a horizontal
plane id found to be 30⁰. Find the height of the tower**

**Ans.) Let AB be the
tower and O be the point of the observation. Then OA = 90 m, and ∠OAB = 90⁰, and ∠AOB = 30⁰**

** AB**

**Let, AB = h meters, Then from right △OAB, we have ------ = tan 30⁰**

** OA**

** h 1**

** => -------- = --------**

** 90 √3**

** 90**

** => h
= -------- = 30√3**

** √3**

**
=> h = (30 X 1.732) m [√3 = 1.732]**

**
=> h =
51.96 m **

**Hence, the height of the tower is 51.96 m (Ans.)**

**Example.2) A kite is flying at a height of 75 m from the level ground,
attached to a string inclined at 60⁰ to the horizontal. Find the length of the
string to the nearest meter.**

**Ans.) Let OX be the horizontal
line and A be the position of the kite with string OA**

**Draw AB ⊥ OX**

**Then, AB = 75 m and ∠BOA = 60⁰**

**From right △OBA, we get**

** AB**

** -------- = sin 60⁰**

** OA**

** 75 √3**

**=> -------- = -------**

** OA 2**

** OA 2**

**=> -------- = -------**

** 75 √3**

** 75 X 2 50 X 3**

**=> OA =
----------- = ----------**

** √3 √3**

**=> OA = (50 X √3) = (50
X 1.732) [where, √3 = 1.732]**

** = 86.6 m **

**Hence, the length of the string to the nearest meter is 87 m. (Ans.)**

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