Trigonometry
Trigonometry is the branch of mathematics that deals with the measurement of angles and helps us study the relationship between the sides and angles of a right-angled triangle. The word trigonometry is a 16th century Latin derivative and the concept was given by the Greek mathematician Hipparchus.
Introduction to Trigonometry
Trigonometry is one of the most important branches of mathematics. The word trigonometry is formed by clubbing words Trigonon and Metron which means triangle and measure respectively. It is the study of the relation between the sides and angles of a right-angled triangle. It thus helps in finding the measure of unknown dimensions of a right-angled triangle using formulas and identities based on this relationship.
You can know about triangles and their different types here. You can also go ahead and explore all important topics in Trigonometry by selecting the topics from this list below:
| Trigonometric Basics | Trigonometric Table |
| Trigonometric Identities | Trigonometric Formulas |
Trigonometric Identities
In Trigonometric Identities, an equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved. In trigonometric identities, you will get to learn more about the Sum and Difference Identities.
For example, sin θ ÷ cos θ = [Opposite/Hypotenuse] ÷ [Adjacent/Hypotenuse] = Opposite/Adjacent = tan θ
Therefore, tan θ = sin θ ÷ cos θ is a trigonometric identity. The three important trigonometric identities are:
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ
Trigonometric Table
The trigonometric table is made up of trigonometric ratios that are interrelated to each other – sine, cosine, tangent, cosecant, secant, cotangent. These ratios, in short, are written as sin, cos, tan, cosec, sec, and cot. You can refer to the trigonometric table chart to know more about these ratios.
Trigonometric Formulae
The complete list of trigonometric formulae involving trigonometric ratios and trigonometric identities is listed for easy access. Here's a list of all the trigonometric formulas for you to learn and revise.
Trigonometric Basics
Trigonometry basics deal with the measurement of angles and problems related to angles. There are three basic functions in trigonometry: sine, cosine, and tangent. These three basic ratios or functions can be used to derive other important trigonometric functions: cotangent, secant, and cosecant are derived. All the important concepts covered under trigonometry are based on these functions. Hence, further, we need to learn these functions and their respective formulas at first to understand trigonometry.
If θ is the angle in a right-angled triangle, formed between the base and hypotenuse, then
- sin θ = Perpendicular/Hypotenuse
- cos θ = Base/Hypotenuse
- tan θ = Perpendicular/Base
In a right-angled triangle,
- perpendicular is the side opposite to the angle θ.
- base is the adjacent side to the angle θ.
- hypotenuse is the side opposite to the right angle
The value of the other three functions: cot, sec, and cosec depend on tan, cos, and sin respectively as given below.
- cot θ = 1/tan θ = Base/Perpendicualr
- sec θ = 1/cos θ = Hypotenuse/Base
- cosec θ = 1/sin θ = Hypotenuse/Perpendicular
Let’s define a few terms that will be used extensively in trigonometry
| Adjacent |
It is the side of the triangle which is adjacent to (or below) angle θ. BC is the adjacent side. |
| Opposite |
It is the side of the triangle which is opposite to angle θ. AB is the opposite side. |
| Hypotenuse |
It is the largest side of the triangle. AC is the hypotenuse. |
| Angle of Elevation |
It is the angle between the horizontal plane and the line of sight from an observer's eye to an object above. θ is the angle of elevation. |
Real-Life Examples of Trigonometry
Trigonometry has many real-life examples used broadly. Let’s get a better idea of trigonometry with an example. A boy is standing near a tree. He looks up at the tree and wonders “How tall is the tree?” The height of the tree can be found without actually measuring it. What we have here is a right-angled triangle, i.e.: a triangle with one of the angles equal to 90 degrees. Trigonometric formulas can be applied to calculate the height of the tree, if the distance between the tree and boy, and the angle formed when the tree is viewed from the ground is given.
It determined using the tangent function, such as tan of angle is equal to the ratio of the height of the tree and the distance. Let us say the angle is θ, then
Tan θ = Height/Distance between object & tree
Distance = Height/Tan θ
Let us assume that distance is 30m and the angle formed is 45 degrees, then
Height = 30/Tan 45°
Since, tan 45° = 1
So, Height = 20 m
The height of the tree can be found out by using basic trigonometric formulae.
Applications of Trigonometry
Throughout history, trigonometry has been applied in areas such as architecture, celestial mechanics, geodesy, surveying, etc. Its applications include in:
- various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, and many more.
- It is also helpful to find the distance of long rivers, measure the height of the mountain, etc.
- spherical trigonometry has been used for locating solar, lunar, and stellar positions.
Solved Examples on Trigonometry
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Example 1: The building is at a distance of 150 feet from point A. Can you calculate the height of this building if tan θ = 4/3?
Solution:
The base and height of the building form a right-angle triangle. Now apply the trigonometric ratio of tanθ to calculate the height of the building.
In Δ ABC, AB = 150 ft, tanθ = (Opposite/Adjacent) = BC/AC
4/3 = (Height/150 ft)
Height = (4×150/3) ft = 200ft∴ The height of the building is 200ft.
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Example 2: A man observed a pole of height 60 ft. According to his measurement, the pole cast a 20 ft long shadow. Find the angle of elevation of the sun from the tip of the shadow?
Solution:
Let x be the angle of elevation of the sun, then
tan x = 60/20 = 3
x = tan-1(3)
or x = 71.56 degreesTherefore, the angle of elevation of the sun is 71.56º.
Practice Questions on Trigonometry
FAQs on Trigonometry
What is Trigonometry?
Trigonometry is the branch of mathematics that deals with the study of the relationship between the sides of a triangle (Right-angled triangle) and its angles.
What are the Basics of Trigonometry?
Trigonometry basics deal with the measurement of angles and problems related to angles. There are six basic trigonometric ratios: sine, cosine, tangent, cosecant, secant and cotangent. All the important concepts covered under trigonometry are based on these trigonometric ratios or functions.
What are the Applications of Trigonometry?
Trigonometry finds applications in different fields in our day-to-day lives. In astronomy, trigonometry helps in determining the distances from the Earth to the planets and stars. It is used in constructing maps in geography and navigation. It can also be used in finding an island's position in relation to the longitudes and latitudes. Even today, some of the technologically advanced methods which are used in engineering and physical sciences are based on the concepts of trigonometry.
How do you do SOH CAH TOA?
We use the "SOH CAH TOA" trick to memorize the relationship between trigonometric ratios easily. To remember them, remember the word "SOHCAHTOA"!
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
What does θ Mean in Trigonometry?
θ in trigonometry is used as a variable to represent a measured angle. It is the angle between the horizontal plane and the line of sight from an observer's eye to an object above. θ can be referred to as the angle of elevation or angle of depression, depending upon the object's position, i.e, when the object is above the horizontal line θ is called the angle of elevation, and for object's position below the horizontal line, it is called the angle of depression.