Students must be well aware of all the important definitions related to trigonometry Height and Distance.

1. Object:
   An object is anything that a person can see. It can be a building, a tree, a pole, or even a person.
2. Viewer:
   The viewer is the person who is looking at the object.
3. Line of Sight:
   This is the imaginary straight line drawn from the eyes of the viewer to the object they are looking at.
4. Horizontal Line:
   A horizontal line is any straight line that is level with the surface of the earth. It is the line that goes straight across from the viewer’s eyes when they are looking straight ahead.
5. Angle of Elevation:
   When the object is located above the eye level of the viewer, the angle between the horizontal line and the line of sight is called the angle of elevation.
6. Angle of Depression:
   When the object is located below the eye level of the viewer, the angle between the horizontal line 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.

Heights and Distances from Trigonometry

In trigonometry, heights and distances refer to the measurement and calculation of vertical heights and horizontal distances between objects or points. Trigonometric functions such as sine, cosine, and tangent play a crucial role in solving problems related to heights and distances. These functions allow us to relate angles, sides, and distances within right-angled triangles.

Consider a typical scenario where a person standing at a certain point wants to measure the height of an object or determine the distance between two objects. By using trigonometric concepts and formulas, we can calculate these measurements accurately.

For example, let's say we want to find the height of a tree. We can choose a point on the ground and measure the distance from that point to the tree. Next, we can measure the angle of elevation from the ground to the top of the tree. Using trigonometric ratios, such as the tangent function, we can set up and solve an equation to find the height of the tree.

Similarly, trigonometry can be used to determine distances between two objects. For instance, if we know the height of an observation tower and the angle of depression from the top of the tower to a point on the ground, we can use trigonometry to calculate the horizontal distance between the tower and the point.

By applying trigonometric principles, including the Pythagorean theorem, trigonometric ratios, and the concept of similar triangles, we can solve a wide range of heights and distances problems. These concepts are widely used in various fields, including architecture, surveying, navigation, and physics.

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 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

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:

\(d=h\left(\cot \alpha -\cot \beta \right)\)

Learn about Value of Tan 180

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.

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.

1. The side opposite to 90° is called hypotenuse of the right-angled triangle
2. 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.
3. Relation between p, b and h is p2 + b2 = h2
4. 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
5. sinθ = 1/cosecθ, cosθ = 1/secθ, tanθ = 1/cotθ
6. cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
7. tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
8. 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!

Properties of Trigonometry: Height and Distance

1. Based on Right-Angled Triangles:
   All height and distance problems in trigonometry are solved using right-angled triangles. The trigonometric ratios (like sine, cosine, and tangent) apply only to right triangles.
2. Use of Angles of Elevation and Depression:
   These problems involve either looking upward (angle of elevation) or downward (angle of depression) from a horizontal line.
3. Trigonometric Ratios Used:
   The three main trigonometric ratios used are:
   • tan(θ) = opposite / adjacent (most commonly used in height-distance)
   • sin(θ) = opposite / hypotenuse
   • cos(θ) = adjacent / hypotenuse
4. Measurements Are in the Same Units:
   Heights, distances, and lengths should all be in the same unit (like meters, feet, etc.) for correct calculations.
5. Assumes Flat Surface (Horizontal Line):
   The horizontal line (like the ground or the viewer’s eye level) is assumed to be perfectly straight and flat.
6. Line of Sight Is a Straight Line:
   The line of sight from the viewer to the object is always a straight line and forms one side of the right triangle.
7. Object and Observer Are in the Same Vertical Plane:
   For calculations to work correctly, the object being viewed and the viewer are assumed to be in the same vertical plane.
8. Heights Can Include Observer’s Eye Level:
   Sometimes, the height of the viewer’s eyes (like 1.5 m from the ground) is included in the total height calculation.
9. Only One Unknown Value at a Time:
   Most problems give enough information so that only one unknown (height or distance) needs to be found.
10. Real-Life Applications:
    These concepts are widely used in real life, such as:

• Finding the height of buildings, trees, towers.
• Measuring the distance of ships from a lighthouse.
• Calculating the angle at which a ladder leans against a wall.

Applications of Trigonometry in Height and Distance

Trigonometry is widely used in calculating unknown heights and distances without directly measuring them. These calculations are mostly based on the angle of elevation and angle of depression.

1. Measuring Height of a Building or Tree

If you know your distance from the building and the angle at which you look up to the top (angle of elevation), you can use trigonometric formulas (like tan θ = height/distance) to find the height.

2. Aviation and Air Navigation

Pilots use the angle of depression to calculate how far they need to descend to reach the runway. Similarly, trigonometry helps in finding the height of clouds or aircraft.

3. Surveying and Civil Engineering

Surveyors use instruments to measure angles and use trigonometry to find the height of mountains, towers, bridges, and other structures.

4. Navigation and Sea Travel

Navigators use trigonometry to find the distance to a lighthouse or a ship from the shore using the height of the lighthouse and the angle of depression.

5. Astronomy and Satellite Tracking

Trigonometry is used to calculate the height or position of stars and satellites above Earth, based on the angle from a point of observation.

6. Shadow Length Estimations

By measuring the shadow and angle of elevation of the sun, one can calculate the height of poles or towers.

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.