Introduction to Trigonometry

Trigonometric Ratios

Six trigonometric ratios for right angle triangle are Sin, Cos, Tan, Cosec, Sec, Cot which stands for Sine, Cosecant, Tangent, Cosecant, Secant respectively. We will learn the formulas for these trigonometric ratios and some funny mnemonics to memorize it.

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Trigonometric Ratios
Trigonometric Identities
Solved Examples
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Trigonometric Ratios in Right Angle Triangle

Trigonometric Ratios are applicable only for a right angle triangle. It is a triangle with specialty, that one angle of the triangle will be of 90o and rest two will be less than 90o. Each side of the right angle triangle has a name:

  • Hypotenuse: Like its name, it is the largest side of the triangle. In other words, it is the largest side opposite to the right angle of the triangle.
  • Base: Side on which right angle triangle stands is known as its base. Any of the two sides other than the hypotenuse can be chosen as the base for performing the calculation.
  • Perpendicular: The side perpendicular to the base the right-angled triangle will be perpendicular to the right-angled triangle.

Learn more about Basics Concepts of Inverse Trigonometric Functions here

Trigonometric Ratios

Before starting with trigonometric ratios, let’s brush up concepts on about what are sine, cosine, and tangent? These are actually the trigonometric ratios. Now the question arises why? Because sine, cosine, and tangent are ratios of sides of a right angle triangle.

Consider a right angle triangle right angle ABC. Let at C be of 90o. Side AB will be hypotenuse then, chose AC as base and BC as perpendicular. Follow the given figure: for angle BAC, value of sinθ = Perpendicular/ hypotenuse = BC/AB

Trigonometric Identities

(Right Angle Triangle ABC)

Concepts of Trigonometric Ratios

Fixing the base and perpendicular can be difficult sometimes. For example in the triangle above,

  • For angle BAC, sinθ1= Perpendicular/ Hypotenuse = BC/AB
  • But for angle ABC, sinθ2= Perpendicular/ Hypotenuse = AC/AB

Confusing, isn’t? To remove this confusion, we will name different sides of the right-angled triangle as adjacent, opposite and hypotenuse.

  • Adjacent: It will be the side adjacent to the angle being considered.
  • Opposite: It will be the side opposite to angle being considered.
  • Hypotenuse: It will be the side opposite to the right angle of the triangle (or the largest side).

Now, formulas for ratios are as follows:

  • sinθ= Perpendicular/ Hypotenuse= Opposite/Hypotenuse
  • cosine or cosθ= Base/ Hypotenuse= Adjacent/Hypotenuse
  • tangent or tanθ= Perpendicular/Base= Opposite/Adjacent.

The reciprocal of sin, cos, and cot can also have names and it’s obvious they are also trigonometric ratios. Those are as follows:

  • cosecθ= Hypotenuse/Perpendicular= Hypotenuse/Opposite
  • seecθ = Hypotenuse/Base =Hypotenuse/ Adjacent
  • cotθ= Base/Perpendicular= Adjacent/Opposite

Understand more about Properties of Inverse Trigonometric Functions here

Trigonometric Ratios Table

trigonometric ratios

source: onlinemath4all.com

Trigonometric Ratios Mnemonics

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, representing the sine, cosine, and tangent (and their corresponding sides) as strings of letters can help us remember them. Consider one famous mnemonic Some People Have, Curly Black Hair Through Proper Brushing.

Here, Some People Have is for

  • Sinθ= Perpendicular/ Hypotenuse.

Curly Black Hair is for

  • Cosθ= Base/ Hypotenuse.

Through Proper Brushing is for

  • Tanθ= Perpendicular/Base

Solved Example for You

Q: For a right angle triangle ABC right angled at C with Hypotenuse=AB = 5cm, Perpendicular=BC =4cm and Base=AC= 3cm Calculate for angle BAC=θ value of sinθ , cosθ and tanθ.

Solution: For right angle triangle ABC,

sinθ =Perpendicular/ Hypotenuse=4/5

cosθ =Base/ Hypotenuse=3/5

tanθ =Perpendicular/Base= 4/3

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