Trigonometry is used in different activities in our day-to-day life. Students need to focus on understanding the basics related to trigonometry to be able to understand the application of trigonometrical functions. Height and distance of different things can be measured with the appropriate use of trigonometrical functions. Students need to be well accustomed to different trigonometrical formulas and functions before they engage in solving sums related to application of trigonometrical functions.

The concept of application of trigonometrical function involves the need of a right angled triangle. Further, it is necessary for the students to be provided with some information like the length of the sides or the angles to be able to calculate the unknown identities. The height of a tower or a tree can be easily determined without climbing over it using trigonometry. Similarly, with the appropriate use of trigonometrical functions one can find the width of a river with the help of certain minor calculations. .

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Students need to work really hard for their board examinations. It is significantly necessary for the students to practice and revise all the topics from time to time. Embibe offers students with a range of study materials which includes PDF of books, solution sets and MCQ mock test papers. Students can follow these solution sets to understand the correct approach to answer the questions appropriately. Further, the mock tests will allow the students to revise all the topics and identify the areas that require further practice. The test papers are prepared considered the marking scheme, exam pattern of CBSE 2022.

History of Application of Trigonometry

The history of applications of trigonometry: Surveyors have used trigonometry for centuries. During one such surveying project of the nineteenth century was the ‘Great Trigonometric Survey of British India, for which the two largest-ever theodolites were built. In \(1852\) during the survey, the mountain with the highest length in the world was discovered. From over \(160{\rm{\;}}\,{\rm{km}},\) the peak was observed from six different stations.

A Theodolite

In \(1856,\) this mountain peak was named after Sir George Everest, who had commissioned and first used the giant theodolites (see the figure above). The theodolites are currently on display in the Museum of the Survey of India in Dehradun.

Application of Trigonometry

Trigonometry is being used for finding the heights and distances of various objects without measuring them. In solving problems of heights and distances, two types of angles are involved:

1. The angle of elevation,
2. The angle of depression
Before knowing these angles, it is necessary to know about the following terms.

• Horizontal plane: A plane parallel to the earth surface is called the horizontal plane.
• Horizontal line: A line drawn parallel to a horizontal plane is called a horizontal line.

Line of Sight

In this figure, the line \(PR\) drawn from the student’s eye to the top of the Qutb Minar is called the line of sight. The student sees at the top of the Qutb Minar. The \(\angle QPR\) so formed by the line of sight with the horizontal, is called the angle of elevation of the top of the Qutb Minar from the student’s eye.

Angle of Elevation

The line of vision is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point considered is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., when we lift our head to look at the object.

Angle of Depression

Now, consider the situation; the girl is sitting on the balcony is looking down at a flowerpot placed on a stair of the temple. In this situation, the line of vision is below the horizontal level.

The angle in such a way formed by the line of sight with the horizontal is called the angle of depression. Therefore, the angle of depression of a point on the object that is viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed.

Note:
\({\rm{The\;angle\;of\;elevation}} = {\rm{the\;angle\;of\;depression}}.\)
The angle of elevation and the angle of depression is measured with respect to a horizontal line.
In solving problems observer is represented by a point, and objects are represented by line segment or point as the case might be.

Applications of Trigonometry Formulas

The height of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

Trigonometric ratios in right triangles: In right triangle \(ABC,\,\angle CAB\) is an acute angle. Observe the position of the side \(\angle A.\) We call it the side perpendicular to angle \(A.\,AC\) is the hypotenuse of the right-angled triangle, and the side \(AB\) is a part of \(\angle A.\) So, we call it the side base to \(\angle A.\)

Trigonometric Ratios in Right Triangles

1. \({\rm{sin\;}}A = \frac{{{\rm{Perpendicular}}}}{{{\rm{Hypotenuse}}}} = \frac{{BC}}{{AC}}\)
2. \({\rm{cos\;}}A = \frac{{{\rm{Base}}}}{{{\rm{Hypotenuse}}}} = \frac{{AB}}{{AC}}\)
3. \({\rm{tan\;}}A = \frac{{{\rm{Perpendicular}}}}{{{\rm{Base}}}} = \frac{{BC}}{{AB}}\)
4. \({\rm{cosec\;}}A = \frac{{{\rm{Hypotenuse}}}}{{{\rm{Perpendicular}}}} = \frac{{AC}}{{BC}}\)
5. \({\rm{sec\;}}A = \frac{{{\rm{Hypotenuse}}}}{{{\rm{Base}}}} = \frac{{AC}}{{AB}}\)
6. \({\rm{cot\;}}A = \frac{{{\rm{Base}}}}{{{\rm{Perpendicular}}}} = \frac{{AB}}{{BC}}\)

NOTE: The value of the trigonometric ratio of an angle do not vary with the length of the side of the triangle if the angle remains the same.

Application of Trigonometry in Real Life

Trigonometry is used in day to day life around us. Trigonometry is among the most ancient subjects studied by scholars everywhere on the planet and was invented for its immense applications in astronomy. Since then, astronomers have used it, for example, to calculate distances of the planets and stars from the earth. Additionally, trigonometry is also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes.

Type I : Problems based on finding the acute angle of a right-angle triangle whose two sides are known.

Working Rule:

1. When base and hypotenuse are known, use \({\rm{cos\theta }} = \frac{b}{h}.\)
2. When hypotenuse and perpendicular are known use \(\sin \theta = \frac{p}{h}.\)
3. When perpendicular and base are known, use \({\rm{tan\theta }} = \frac{p}{b}.\)

Type II: Problems based on finding a side of a right-angled triangle when one acute angle and one of the two remaining sides are known.

Working Rule: Use the following results, whichever is required:

1. When one of the hypotenuse and perpendicular is known, and the other is to be determined, use \({\rm{sin\theta }} = \frac{p}{h}.\)
2. When one of hypotenuse and base is known, and the other is to be determined, use \({\rm{cos\theta }} = \frac{b}{h}.\)
3. When one of perpendicular and base is known, and the other is to be determined using \({\rm{tan\theta }} = \frac{p}{b}.\)

Solved Examples on Applications of Trigonometry

Q.1. A vertical tower is \(3\sqrt 3 \,{\rm{m}}\) high, and the length of its shadow is \(3\,{\rm{m}}.\) Find the angle of elevation of the source of light.
Ans:

Given, \(PQ = 3\sqrt 3 \,{\rm{m}},{\rm{\;QR}} = 3{\rm{\;}}\,{\rm{m}}\)
Let \(PQ\) be the vertical tower, \(S\) be the source of light, and \(QR\) be the shadow of tower \(PQ.\)
Let angle of elevation of \(S,\) i.e., \(\angle PRQ = {\rm{\theta }}.\)
Now in right-angled \(\Delta PQR,\)
\(\tan {\rm{\theta }} = \frac{{PQ}}{{QR}} = \frac{{3\sqrt 3 }}{3} = \sqrt 3 = \tan 60^\circ \)
Hence angle of elevation of the light source is \(60^\circ \)

Q.2. In \(\Delta XYZ,\,\angle X = 90^\circ ,\,XY = 90\,{\rm{cm}}\) and \(AC = 3\sqrt 3 \,{\rm{cm}},\) then find \(\angle Y.\)
Ans:

Let in \(\Delta XYZ,\,\angle X = 90^\circ \)
\(XY = 9\,{\rm{cm}}\) and \(XZ = 3\sqrt 3 {\rm{\;cm}}{\rm{.}}\) Let \(\angle Y = {\rm{\theta }}.\)
In right angled \(\Delta XYZ,\)
\({\rm{tan\theta }} = \frac{{XZ}}{{XY}} \Rightarrow {\rm{tan\theta }} = \frac{{3\sqrt 3 }}{9}\)
\(\Rightarrow {\rm{tan\theta }} = \frac{{\sqrt 3 }}{3} \Rightarrow {\rm{tan\theta }} = \frac{1}{{\sqrt 3 }}\)
\(\Rightarrow {\rm{\theta }} = 30^\circ \)
Hence \(\angle Y = 30^\circ.\)

Q.3. An aeroplane at an altitude of \(300\,{\rm{m}}\) observes the angles of depression of opposite points on the two banks of a river to be \(30^\circ\) and \(45^\circ.\) Find the width of the river.
Ans: Let \(A\) be the position of the aeroplane and let \(P\) and \(Q\) be two points on the two banks of a river such that the angles of depression at \(P\) and \(Q\) are \(30^\circ \) and \(45^\circ, \) respectively.

In \(\Delta AMP,\) we have
\({\rm{tan}}30^\circ = \frac{{AM}}{{PM}}\)
\(\Rightarrow \frac{1}{{\sqrt 3 }} = \frac{{300}}{{PM}}\)
\(\Rightarrow PM = 300\sqrt 3 \)
In \(\Delta AMQ,\) we have
\({\rm{tan}}45^\circ = \frac{{AM}}{{MQ}}\)
\(\Rightarrow 1 = \frac{{300}}{{MQ}}\)
\(\Rightarrow MQ = 300\)
Now, \(PQ = PM + MQ = 300\sqrt 3 + 300\)
\(PQ = 300\left( {\sqrt 3 + 1} \right)\;{\rm{m}}\)
Hence, the measure of \(PQ\) is \(300\left( {\sqrt 3 + 1} \right)\;{\rm{m}}\)

Q.4. A tree \(10\,{\rm{m}}\) high is broken by the wind in such a way that its top touches the ground and makes an angle \(30^\circ \) with the ground. At what height from the bottom, the tree is broken by the wind?
Ans:

Let \(PQ\) be the tree of height \(10\,{\rm{m}}.\) Suppose the tree is broken by the wind at point \(R,\) and the part \(RQ\) assumes the position \(RO\) assumes the position \(O.\)
Let \(PR = a.\) Then, \(RO = RQ = 10 – a.\) It is given that \(\angle POR = 30^\circ \)
In \(\Delta POR,\) we have \(\sin 30^\circ = \frac{{PR}}{{OR}} = \frac{a}{{10 – a}}\)
\(\Rightarrow \frac{1}{2} = \frac{a}{{10 – a}}\)
\(\Rightarrow 2a = 10 – a\)
\(\Rightarrow a = \frac{{10}}{3}\;{\rm{m}} = 3.33\,{\rm{m}}\)
Hence, the tree is broken at a height of \(3.33\,{\rm{m}}\) from the ground.

Q.5. A circus artist is climbing a \(10\,{\rm{m}}\) long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is \(60^\circ.\)
Ans: Let \(PQ\) be the vertical pole and \(PR\) be the \(20\,{\rm{m}}\) long rope such that one end is tied from the top of the vertical pole \(PQ\) and the other end \(R\) and the other end \(R\) on the ground.

In \(\Delta PQR,\) we have
Therefore, \(\sin 60^\circ = \frac{{PQ}}{{PR}} = \frac{{PQ}}{{10}}\)
\(\Rightarrow \frac{{\sqrt 3 }}{2} = \frac{{PQ}}{{10}}\)
\(\Rightarrow PQ = \frac{{10\sqrt 3 }}{2} = 5\sqrt 3 \,{\rm{m}}\)
Hence, the height of the pole is \(5\sqrt 3 \,{\rm{m}}{\rm{.}}\)

Summary

In this article, we learnt about, history of Applications of Trigonometry, Applications of Trigonometry, Definition of Angle of Elevation, Definition of Angle of Depression, Applications of Trigonometry Formulas, Application of Trigonometry in real life. At the end of this article we have discussed few examples for a better understanding of the topic.

Frequently Asked Questions (FAQs) on Applications of Trigonometry

Frequently asked questions related to application of trigonometrical functions are listed as follows:

Q.1. How to implement applications of Trigonometry?
Ans: Students need to have complete knowledge of both trigonometrical functions as well as formulas to be able to apply trigonometrical functions in different problem sums.

Q.2. What are examples of applications of Trigonometry in real life?
Ans: a. Trigonometry is used to measure the height of a building, towers or mountains.
b. Trigonometry can be used to roof a house, make the roof inclined ( in the case of single individual bungalows), the height of the top in buildings, etc.

Q.3. What is angle of depression and elevation?
Ans: You can refer to the article above to get all the details related to angle of depression and elevation.

Q.4. Who is the father of trigonometry?
Ans: The word ‘trigonometry’ is derived from the Greek words ‘tri’(meaning three), ‘gon’  (meaning sides)  and  ‘metron’  (meaning measure).  In fact, trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on trigonometry was recorded in Egypt and Babylon. The first trigonometric table was apparently compiled by Hipparchus, who is consequently now known as the father of trigonometry.

Q.5. What are the uses of applications of Trigonometry?
Ans: a. It is used in oceanography in calculating the height of tides in oceans.
b. The sine and cosine functions are fundamental to the theory of periodic functions, those that describe sound and light waves.
c. It is used in the naval and aviation industries.
d. It is used in the creation of maps.
e. Trigonometry has its applications in satellite systems.

We hope this detailed article on Applications of Trigonometry is helpful to you.