Trigonometry Height and Distance with Definitions, Formulas and Solved Examples
Saurav Sinha
| Updated: Jan 31, 2023 17:01 IST
Students must be well aware of all the important definitions related to trigonometry Height and Distance.
(a) Object: A thing or a person seen by someone can be said to be an object with respect to the viewer.
(b) Viewer: The person who sees the object.
(c) Line of Sight: An imaginary line joining the object and viewer’s eyes is called the line of sight.
(d) Horizontal Line: All the lines parallel to the surface of the earth are called horizontal lines.
(e) Angle of Inclination: When an object is present above the eye’s level of the viewer, then the angle between horizontal (eye level) and the line of sight is called the angle of elevation.
(f) Angle of Depression: When an object is present below the eye’s level of the viewer, then the angle between horizontal (eye level) and the line of sight is called the angle of depression.
Students can find the pictorial representation of all the common terminologies and definitions of height and distance mentioned below.
What is Height and Distance?
Height can be defined as the measurement of an object in the vertical direction and distance can be defined as the measurement of an object from a particular point in the horizontal direction.
Usually, all government competitive examinations ask questions based on Trigonometry under Quantitative Aptitude.
How to Find Height and Distance?
Trigonometric ratios are used to find the heights and distances of different objects. For instance, if a man is looking at the top of a lamppost. Let the following figure represent different points related to height and distance.
In the above figure, we can see that AB is the horizontal level, which is parallel to the ground and passes through the eye of the observer. AC is the line of sight for drawn between the eye of the observer and the top of the lamppost. Angle A formed between the line of sight and the horizontal level can be termed as the angle of elevation.
In order to find the angle of elevation, we can use the trigonometry ratios in the right triangle ABC when the distance between the observer and the lamppost and the height of the lamppost is already given.
We can consider the following table when we have to find the angle of elevation using the different height and distances formula
| Angles | 0-degree | 30-degree | 45-degree | 60-degree | 90-degree |
| Sin C | 0 | 1/2 | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | 1 |
| Cos C | 1 | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | 1/2 | 0 |
| Tan C | 0 | \(\frac{1}{\sqrt{3}}\) | 1 | \(\sqrt{3}\) | Not defined |
| Cot C | Not defined | \(\sqrt{3}\) | 1 | \(\frac{1}{\sqrt{3}}\) | 0 |
| Cosec C | Not defined | 2 | \(\sqrt{2}\) | \(\frac{2}{\sqrt{3}}\) | 1 |
| Sec C | 1 | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{2}\) | 2 | Not defined |
Trigonometry Height and Distance Formula
To understand the trigonometry height and distance formula, let us understand this using example:
(i) Consider the following figure:
We know that for a right-angle triangle, the common trigonometric ratios are written as:
\(\sin \theta =\frac{p}{h}\)
\(\cos \theta =\frac{b}{h}\)
\(\tan \theta =\frac{p}{b}\)
Here, ‘p’ is the perpendicular, ‘h’ is the hypotenuse and ‘b’ is the base of the right-angled triangle.
(ii) Consider the following figure:
In the given figure, if BD: DC = m: n and \(\angle BAD=\alpha ,\ \angle CAD=\beta ,\ \angle ADC=\theta \),
Then, \(\left(m+n\right)\cot \theta =m\cot \alpha -n\cot \beta \)
(iii) Consider the following figure:
In the given figure, if DE is parallel to AB, then, \(\frac{AB}{DE}=\frac{BC}{DC}\)
(iv) Consider the following figure:
In the above figure, we can calculate the value of distance ‘d’ by:
\(d=h\left(\cot \alpha -\cot \beta \right)\)
Trigonometry and Angle
Consider a revolving line OP. Suppose that it revolves around in an anticlockwise direction starting from its initial position OX. The angle is defined as the amount of revolution that the revolving line makes with its initial position.
The angle is taken positive if it is traced by the revolving line in the anticlockwise direction and is taken negative if it is covered in the clockwise direction.
One radian is the angle subtended at the centre of a circle by an arc of the circle, the whole length is equal to the radius of the circle.
Trigonometric Ratios of angles ranging from 0 to 180°
Candidates can find the trigonometric ratio chart below.
| Angle | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | – 1/2 | -1/√2 | -√3/2 | -1 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ | -√3 | -1 | -1/√3 | 0 |
Four Quadrants and ASTC Rule
Candidates should remember the below-mentioned rule to find the sign and magnitude of any trigonometry angle quickly.
- In the first quadrant, all trigonometric ratios are positive.
- In the second quadrant, only sin θ and cosec θ are positive
- In the third quadrant, only tan θ and cot θ are positive.
- In the fourth quadrant, only cos θ and sec θ are positive.
Note: Remember these as, Add Sugar To Coffee or After School To College.
Important Relations in Trigonometry Height and Distance
Candidates need to be well aware of the important relationships related to various items in Height and Distance math to solve the questions successfully.
- The side opposite to 90° is called hypotenuse of the right-angled triangle
- The side opposite to angle θ is called a perpendicular to the right-angled triangle and the third side is called the base of the right-angled triangle.
- Relation between p, b and h is p2 + b2 = h2
- sinθ = p/h, cosθ = b/h, tanθ = p/b, cotθ = b/p, secθ = h/b, and cosecθ = h/p, where p, b h are the perpendicular, base and hypotenuse of the right angles triangle
- sinθ = 1/cosecθ, cosθ = 1/secθ, tanθ = 1/cotθ
- cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
- tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
- sin (90°- θ) = cosθ, cos (90°- θ) = sinθ, tan (90°- θ) = cotθ, cot (90°- θ) = tanθ, cosec (90°- θ) = secθ, sec (90°- θ) = cosecθ
Height and Distance Tips and Tricks
Candidates can find different tips and tricks below for solving the height and distance questions related to Trigonometry Height and Distance.
Tip # 1: The angle of elevation is always equal to the angle of depression.
Tip # 2:The angle of elevation of the sun at a particular time from a particular region on earth is always the same.
When you’ve finished with Height and Distance math, you can read about Percentages concepts in-depth here!
Height and Distance Questions with Solutions
Question 1: From the top of a building a person looks at a parked car and the angle of depression is θ, if the distance between the foot of the building and the parked car is X unit, in this case, the height of the building will be?
Solution: As the angle of depression is equal to the angle of inclination,
So, in this case tanθ = height of building/X
Therefore, height of building = X tanθ
Question 2: If the height of two towers are X unit and Y unit respectively and the length of shadow of the first tower is Z unit, then the length of shadow of the second tower will be?
Solution: From the help of the second memory tip the angle of elevation in both the figures is the same so both the triangles are similar to each other.
So, X / Z = Y / ?
Hence, ? = ZY / X
Question 3: The angle of elevation of the top of a tower from two points from the ground is complementary and the distance between these points from the foot of the tower are X and Y, then the height of the tower is?
Solution: From figure,
tanθ = TF/X—— (i)
and tan(90°- θ) = TF/Y cotθ = TF/Y ————— (ii)
from (i) and (ii), we get tanθ × cotθ = TF/X × TF/Y ⇒ 1 = (TF)2/XY ⇒ TF = √XY
Therefore, the height of tower will be √XY
Also check Profit and Loss concepts here once you are through with Height and Distance concepts!
Exams where Height and Distance is Part of Syllabus
Height and Distance questions come up often in various prestigious government exams some of them are as follows.
- SBI PO, SBI Clerk, IBPS PO, IBPS Clerk
- SSC CGL, SSC CHSL, SSC MTS
- LIC AAO, LIC ADO
- RRB NTPC, RRB ALP
- UPSC
- MPSC
- KPSC
- BPSC
- WBPSC
- Other State Level Recruitment Examinations
We hope you found this article regarding height and distance was informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation.
| If you are checking Height and Distance article, also check the related maths articles in the table below: | |
| Mensuration 2D | Mensuration 3D |
| Data Interpretation | Mixture and Alligation |
| Time and Work | Double Line Graph |
Height and Distance FAQs
Q.1 What is Height and Distance?
Ans.1 Height can be defined as the measurement of an object in the vertical direction and distance can be defined as the measurement of an object from a particular point in the horizontal direction.
Q.2 Where can I get the important relations of Height and Distance?
Ans.2 Candidates can find some of the important relations used in Height and Distance based questions from above.
Q.3 In which exam questions from Height and Distance come up?
Ans.3 Height and Distance comes in various government competitive examinations such as SBI PO, SBI Clerk, IBPS PO, IBPS Clerk, SSC CGL, SSC CHSL, SSC MTS, LIC AAO, LIC ADO, RRB NTPC, RRB ALP, UPSC, MPSC, KPSC, BPSC, WBPSC.
Q.4 How to find the height of an object using trigonometry?
Ans.4
The height of a given object can be calculated using trigonometric ratios. We know that:
\(\tan \theta =\frac{p}{b}\)
Where, ‘p’ is the perpendicular or the height, and ‘b’ is the base of the right-angled< triangle.
In order to find the height of the given object, we need to know the angle of elevation and the distance of the object from the observer. We can equate it in the above formula to find the value of the perpendicular.
