Heights and Distances

In mathematics, trigonometry is one of the most important branches. Thus, the application of trigonometry includes, measuring the heights of towers, determining the distance of the object in the sea from the shore, finding the distance between celestial bodies, etc. Mainly, we use trigonometry to measure heights and distances. Therefore, we use trigonometric ratios to measure heights and distances of different objects.

Heights and Distances

Before we study height and distances we need to understand some terms.

1. Angle

An angle is a measure of rotation of a ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. Vertex is the point of rotation.

If the ray rotates in an anticlockwise direction, the angle is said to be a positive angle and if the ray rotates in the clockwise direction, the angle is a negative angle.

 

 

Source: clipart-library.com

 

Degree Measure

If a rotation from the initial side to terminal side is 360^th of a revolution, the angle is said to have a measure of one degree, written as 1°.

Radian Measure

The angle subtended at the centre by an arc of length 1 unit in a unit circle whose radius is 1 unit is said to have a measure of 1 radian. It is another way of measuring any angle.

In a circle of radius r, subtends an angle θ radian at the centre and an arc of length l,

we have l= \(\frac{θ}{r}\)

Relation Between Degrees and Radians : 2π radian = 360° or  π radians = 180° 

Degree	30o	45o	60o	90o	180o	270o	360o
Radian	\(\frac{π}{6}\)	\(\frac{π}{4}\)	\(\frac{π}{3}\)	\(\frac{π}{2}\)	π	\(\frac{3π}{2}\)	2π

Trigonometric Ratios

In a right-angled  \(\Delta\)BAO, where BOA = \(\Theta\),

                                                                                                                                      Source: clipart-library.com

i.   sin \(\Theta\) = \(\frac{Perpendicular}{Hypotenuse}\) = \(\frac{AB}{OB}\)

ii.  cos \(\Theta\) = \(\frac{Base}{Hypotenuse}\) = \(\frac{OA}{OB}\)

iii.  tan \(\Theta\) = \(\frac{Perpendicular}{Base}\) = \(\frac{AB}{OA}\)

iv.  cosec \(\Theta\) = \(\frac{1}{sinθ}\) = \(\frac{OB}{AB}\)

v.  sec \(\Theta\) = \(\frac{1}{cosθ}\) = \(\frac{OB}{OA}\)

vi.  cot \(\Theta\) = \(\frac{1}{tanθ}\) = \(\frac{OA}{AB}\)

Trigonometrical Identities

1. sin²\(\Theta\) + cos²\(\Theta\)  = 1
2. 1+ tan²\(\Theta\) = sec²\(\Theta\)
3. 1+ cot²\(\Theta\) = cosec²\(\Theta\)

Values of Trigonometric-Ratios

\(\Theta\)	0°	\(\frac{π}{6}\)    30°	\(\frac{π}{4}\)     45°	\(\frac{π}{3}\)   60°	\(\frac{π}{2}\)   90°
sin\(\Theta\)	0	\(\frac{1}{2}\)	\(\frac{1}{2}\)	\(\frac{3}{2}\)	1
cos\(\Theta\)	1	\(\frac{3}{2}\)	\(\frac{1}{2}\)	\(\frac{1}{2}\)	0
tan\(\Theta\)	0	\(\frac{1}{3}\)	1	3	not defined

Line of Sight

The line which is drawn from the eyes of the observer to the point being viewed on the object is called the line of sight.

The angle of Elevation

It is the angle which is formed by the line of sight with the horizontal level and the point on the object (above horizontal level) viewed by the observer.

The angle of elevation of point P from O = ∠AOP.

The angle of Depression

It is the angle which is formed by the line of sight with the horizontal level and the point on the object (below horizontal level) viewed by the observer.

The angle of Depression = ∠AOP

Points to remember :

• The angle of elevation is numerically equal to the angle of depression. They are always acute angles.
• Both the angles are measured with the horizontal level.
• To determine the unknown side of the right angled triangle where one side is known and the acute angle is known. So we combine both the sides with trigonometric ratios of the triangle. i.e.

\(\frac{known side}{unknown side}\) = \(\frac{given side}{required side}\) = suitable t – ratio of Δ

Solved on Heights and Distances

Q 1. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60⁰. When he moves 40 m away from the bank, the angle of elevation measured is 30⁰. Find the width of the river and the height of the tree.

Solution:- Let the height of the tree be y and width of the river be x and CD = 40 m

In ΔABD, tan 30⁰ = \(\frac{AB}{BD}\)

\(\frac{1}{\(\sqrt{3}\)}\) = \(\frac{y}{x + 40}\) ⇒ x+40 = y\(\sqrt{3}\)………..(i)

In ΔABC,

tan 60⁰ = \(\frac{AB}{BC}\)

\(\sqrt{3}\) = \(\frac{y}{x}\) ⇒ y =  x\(\sqrt{3}\)……………(ii)

Putting the value of y from (ii) to (i)

x+40= [x\(\sqrt{3}\)] \(\sqrt{3}\) = 3x

40 = 2x

⇒ x= 20

y = x\(\sqrt{3}\) = 20\(\sqrt{3}\)

Height of the tree = 20\(\sqrt{3}\) m

Width of the river = 20 m.