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Chapter 8 – Introduction To Trigonometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You might have come across trigonometry questions during your maths class. But do you know trigonometry is also used in the measurement of height and distance? Trigonometry is considered one of the most significant and crucial branches of mathematics. Hipparchus, a Greek mathematician, gave the concept of trigonometry. As you know, trigonometry aids in findings the angles and missing sides of a triangle through trigonometric ratios, which we will discuss in the article. Here you will comprehend trigonometric ratios and their types and trigonometric identities. We will try to explain trigonometry in easy language and a simple manner. The topics are easier once you understand them. We hope this article helps you understand the concepts easily.

The ratio of sides of a right-angled triangle concerning any of its acute angles is known as the trigonometric ratios of that particular angle. 

There are mainly six types of trigonometric ratios. The following are the six types of trigonometric ratios:-

Sin θOpposite side to θ/Hypotenuse
Cos θAdjacent side to θ/Hypotenuse
Tan θOpposite side/Adjacent side & Sin θ/ cos θ
Cot θAdjacent side/Opposite side & 1/tan θ
Sec θHypotenuse/ Adjacent side & 1/cos θ
Cosec θHypotenuse/Opposite side & 1/sin θ

Range of trigonometric ratios from 0 to 90 degrees

For 0°≤θ≤90°,

  • 0≤sinθ≤1
  • 0≤cosθ≤1
  • 0≤tanθ<∞
  • 1≤secθ<∞
  • 0≤cotθ<∞
  • 1≤cosecθ<∞

tanθ and secθ are not defined at 90°.

cotθ and cosecθ are not defined at 0°.

Variation of trigonometric ratio from 0 to 90 degrees

As θ increases from 0° to 90°

  • sin θ increases from 0 to 1
  • cos θ decreases from 1 to 0
  • tan θ increases from 0 to ∞
  • cosec θ decreases from ∞  1
  • sec θ increases from 1 to ∞
  • cot θ decreases from ∞  0

Standard values of Trigonometric Ratios

A0°30°45°60°90°
sin A01/2 1/√2√3/21
cos A1√3/2 1/√21/20
tan A01/√31√3Not defined
cosec ANot defined2√2 2/√31
sec A12/√3√22Not defined
cot ANot defined√311/√30

Complementary angles are those sets of two angles whose sum is equal to 90 o. Let us understand it through an example. Let say there are two angles ∠A and ∠B. They will be called complementary if,

∠A + ∠B = 90 o

Here,  ∠A is complementary to ∠B and vice versa. 

The lengths of sides of a right-angle triangle’s relationship with the acute angle are shown by trigonometric ratios.  The following relation is established for trigonometric ratios of complementary angles:-

  • sin (90°− θ) = cos θ
  • cos (90°− θ) = sin θ
  • tan (90°− θ) = cot θ
  • cot (90°− θ) = tan θ
  • cosec (90°− θ) = sec θ
  • sec (90°− θ) – cosec θ

So it is concluded that:-

  • Sin of an angle = Cos of its complementary angle
  • Cos of an angle = Sin of its complementary angle
  • Tan of an angle = Cot of its complementary angle

Trigonometric identities are helpful and useful when trigonometric functions are included in an expression or an equation. However, these identities have certain functions of one or more angles. 

Trigonometric Identities List

Reciprocal Identities 

  • Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
  • Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
  • Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

Pythagorean Identities

  • sin2 a + cos2 a = 1
  • 1+tan2 a = sec2 a
  • Cosec2 a = 1 + cot2 a

Ratio Identities

  • Tan θ = Sin θ/Cos θ
  • Cot θ = Cos θ/Sin θ

Opposite Angle Identities

  • Sin (-θ) = – Sin θ
  • Cos (-θ) = Cos θ
  • Tan (-θ) = – Tan θ
  • Cot (-θ) = – Cot θ
  • Sec (-θ) = Sec θ
  • Csc (-θ) = – Csc θ

What are the basics of trigonometry?

Answer.  There are three basic functions in trigonometry that are sine, cosine, and tangent. With these functions’ help, the other three functions are derived: cotangent, secant, and cosecant. Since all the trigonometrical concepts revolve around these functions, it is important to understand these functions first. 

What do you mean by trigonometry?

Answer. Trigonometry deals with the relationship between the sides of a triangle with its angle. To know more about it, you can visit MSVgo official site or download the MSVgo app.

What is the importance of trigonometry?

Answer.  The importance of trigonometry are as follows:

  • It helps solve complex design problems.
  • It is used in physics to measure dynamics, kinematics, and calculating vectors.
  • It helps manipulate equations. 

Who is the father of trigonometry?

Answer. The concept of trigonometry was founded by Hipparchus, a Greek astrologer, geographer, and mathematician.

What is trigonometry with an example?

Answer.  There are various trigonometry examples, such as it is used to find the distance of long rivers and measure the height of buildings or mountains.

There are various applications of trigonometry, such as in oceanography, seismology, meteorology. It is also helpful in measuring the height of the mountain as well. Therefore the proper understanding of trigonometry is paramount. We help you understand trigonometry basics first, with in-depth concept notes and explanatory video on the MSVgo app. The MSVgo philosophy is to enable a core understanding of any concept. MSVgo app has a video library that explains concepts with examples or explanatory visualizations or animation. To learn more about it, check out the MSVgo app and their official site. Stay tuned with the MSVgo app and cheerful learning!

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