Trigonometry Heights And Distances

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The word "Trigonometry" is derived from the two Greek words meaning measurements or solution of triangles. Trigonometry which is a branch of mathematics that deals with the ratio between the sides of a right triangle and its angles.
Trigonometry is the study about relationships between the sides and angles of a triangle.
Trigonometry is used in surveying and to determine Heights and Distances, in navigation it is to determine the location and the distances, and in the fields like nondestructive testing for determining things such as the angle for reflection or refraction of an ultrasound wave.

IMPORTANT TERMS USED IN HEIGHTS AND DISTANCE:

Horizontal ray: A ray parallel to the surface of the earth emerging from the eye of the observer is known as horizontal ray.

Ray of vision: The ray from the eye of the observer towards the object under observation is known as the ray of vision or ray of sight.

Angle of Elevation: If the object under observation is above the horizontal ray passing through the point of observation, the measure of the angle formed by the horizontal ray and the ...
... ray of vision is known as angle of elevation.

.

ray

Angle of depression: If the object under observation is below the horizontal ray passing through the point of observation, the measure of the angle formed by the horizontal ray and the ray of vision is known as angle of depression.

vertical

Practical examples on trigonometry- heights and distances:

Example on heights and distances

1)From a cliff 150m above the shore line the angle of depression of a ship is 19 0 30 '. Find the distance from the ship to a point on the shore directly below the observer.

Solution heights and distances:- Let OB be the cliff of height 150 m , A be the position of the ship the angle of depression of the ship is 19 0 30' , ABC = 19 0 30' .

ABC = 19 0 30' = OAB = ABC = 19 0 30'

trignometry

In Right angled triangle Δ AOB, we want to find )A = adjacent side , OB = opposite side = 150m

Cot ( 19 0 30' ) = (OA)/(OB) = (OA)/(150)

OA = 150 Cot ( 19 0 30' ) = 150 tan ( 90 - 19 0 30' )

OA = 150 * tan ( 70 0 30') = 150 ( 2.8239) = 423.59 m

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Example 2 on heights and distances

A person standing on the bank of a river observers that the angle of elevation of the top of a tree standing on the opposite bank is 60 0 . When he was 40m away from the bank he finds that the able of elevation to be 30 0 . Find the (i) height of the tree, (ii) The width of the river, correct to two decimal places.

Solution heights and distances: Let CD is the tree , CD = h m high

BC is the river , BC = x width of the river

The observer is at A , From figure DAC = 30 0 and DBC = 60 0

trignometry

In right angled ΔBCD , x is the adjacent side , h is the opposite side .

tan 60 0 = (DC)/(BC) = √3 =(DC)/(BC)

= √3 = (h)/(x) = √3 x = h ---------------------------(1)

From right angled ΔACD , tan 30 0 = (DC)/(AC) AC = 40 +x

(1)/(3 ^ (1/2)) = (h)/(x+40) = x +40 = √3h --------------------(2)

Plugging in value of h from 1 in 2

x +40 = √ 3 ( x √3)

x +40 = 3x

-x -x

40 = 2x divide by 2 , x = 20

h = √3 x = 1.732 (20) = 34.64 [ √3 = 1.732 ]

The height of the tree = 34.64 m , width of the river = 20 m

More Examples on heights and distances

Example 3 on heights and distances

An airplane at an altitude of 500 m observes the angle of depression of opposite point on the two banks of a river to be 32 0 28’ and 40 0 12’ respectively . Find in meters the width of the river.

Solution to heights and distances : A is the position of the airplane , DC is the river .

From figure , DAP = 32 0 18’ , ADB = 32 0 18’ and

CAQ = 40 0 12’ , ACB = 40 0 12’ (alternate interior angles )

trignometry

In right angled ΔDBA In right angled Δ ABC

Cot ( 32 0 18 ‘)

= DB / AB Cot (40 0 12’) = BC / AB

DB = AB cot ( 32 0 18’) BC = 500 cot (40 0 12’)

DB = 500 tan ( 90 0 – 32 0 18’) BC = 500 tan (90 0 – 40 0 12’)

= 500 tan (57 0 42 ‘)

= 500 tan (49 0 48’)

= 500 (1.5818)

= 500(1.1833)

DB = 790.9

BC = 591.65

Width of the river DC = DB +BC = 790.9 +591.65 = 1382.55m

Trigonometry deals with the angle and side of triangles. Unidentified angles or lengths are calculated by using trigonometrical ratios such as sine, cosine, and tangent. Learn the initial applications of trigonometry in the fields of examples like navigation, astronomy and surveying, and generally involved working out a remote distance such as the distance of the Earth from the Moon.

Let us learn some example problems in trigonometry.

Learn trigonometry formula and functions with examples:

Sin A = opposite / Hypotenuse

Cos A = Adjacent / Hypotenuse

Tan A = opposite / Adjacent

Learn more on about Numerical Methods for Partial Differential Equations and its Examples. Between, if you have problem on these topics Expanding Logarithms Examples, Please share your comments.