Introduction to Trigonometry

Trigonometric Ratios

The six trigonometric ratios of a right angle triangle are Sin, Cos, Tan, Cosec, Sec and Cot. They stand for Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent respectively. In the following section, we will learn the formulas for these trigonometric ratios. We will also learn some funny mnemonics to memorize it.

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Trigonometric Ratios in Right Angle Triangle

Trigonometric Ratios are applicable only for a right-angle triangle. A right-angle triangle is a special triangle in which one angle is 90o and the other two are less than 90o. Furthermore, each side of the right angle triangle has a name.

  • Hypotenuse: It is the largest side of the triangle. Also, it is opposite the right angle of the triangle.
  • Base: The side on which the right angle triangle stands is known as its base. Moreover, any of the two sides other than the hypotenuse can be chosen as the base for performing the calculation.
  • Perpendicular: It is the side perpendicular to the base of the right-angled triangle.

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trigonometry cheat sheet

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Trigonometric Ratios Definition

Trigonometric ratios are the ratios of sides of a right-angle triangle. The most common trigonometric ratios are sine, cosine, and tangent.

Consider a right-angle triangle ABC, right-angled at C. In that case, side AB will be the hypotenuse. Also, if we chose AC as the base and BC as the perpendicular. Then, for ∠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 ∠BAC, sinθ1= Perpendicular/ Hypotenuse = BC/AB
  • But for ∠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 is the side adjacent to the angle being considered.
  • Opposite: It is the side opposite to angle being considered.
  • Hypotenuse: It is the side opposite to the right angle of the triangle (or the largest side).

Now, formulas for ratios are as follows:

  • sine or 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 tan can also have names. Also, it’s obvious that they are trigonometric ratios. They are as follows:

  • cosecant or cosecθ= Hypotenuse/Perpendicular= Hypotenuse/Opposite
  • secant or secθ = Hypotenuse/Base =Hypotenuse/ Adjacent
  • cotangent or cotθ= Base/Perpendicular= Adjacent/Opposite

What are the Properties of Inverse Trigonometric Functions?

Trigonometric Ratios Table

trigonometric ratios


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 Examples for You

Question 1: Consider a right angle triangle ABC right angled at C. If the hypotenuse=AB = 5cm, perpendicular=BC =4cm and base=AC= 3cm. Then, for ∠BAC=θ, calculate the value of sinθ, cosθ and tanθ.

Answer: For right angle triangle ABC,

sinθ =Perpendicular/ Hypotenuse=4/5

cosθ =Base/ Hypotenuse=3/5

tanθ =Perpendicular/Base= 4/3

Question 2: What are the 3 trigonometric ratios?

Answer: The 3 trigonometric ratios are sine, cosine and tangent. We can find out the sine (or cosine or tangent) of either of the known- 90° angles.

Question 3: What is the use of trigonometric ratios in right angle triangle?

Answer: Trigonometric ratios apply to a right angle triangle only. It is a special triangle in which one angle is 90° and the other two are less than 90°. Also, each side of the triangle has a name. They are hypotenuse, perpendicular and base.

Question 4: What is the tangent ratio?

Answer: It is a tool we use with right triangles. It lets us find the lengths of the sides when the degrees of its angles are given. Also, we can use it to find out the degrees of its angle when the lengths of two of the sides are given.

Question 5: What is a hypotenuse?

Answer: A hypotenuse is the largest side of the triangle. In other words, it refers to the largest side opposite to the right angle.

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