Trigonometric Ratios

What are Trigonometric Ratios?

  • Trigonometric ratios in trigonometry are derived from the three sides of a right- angled triangle basically the hypotenuse, the base(adjacent) and the perpendicular(opposite).

  • These trigonometric formulas and trigonometric identities are used widely in all sciences related to geometry, mechanics and many others.


Trigonometric Ratios Definition

  • Trigonometric ratios help us to find missing angles and missing sides of a triangle.

  • To be more specific, they are used in right- angled triangles, the triangles with one angle equal to 90 degrees.


In case you might not know what, a right-angled triangle is, here’s a little information!


Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle . The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.

                   

The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). The side that is opposite to the angle θ is known as the opposite(O). And the side which lies next to the angle  is known as the Adjacent(A)

The Pythagoras theorem states that,


In a right-angle triangle, (Opposite)2+(Adjacent)2= (Hypotenuse)2


According to trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.


Here they are!


What are Trigonometric Ratios?

Name

Abbreviation

Relationship

Sine

Sin

\[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]

Cosine

Cos

\[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

Tangent

Tan

\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]


  1. Sine: Sine of an angle is defined as the ratio of the side opposite to the angle (θ) to the hypotenuse (longest side) in the triangle.

  2. Cosine: Cosine of an angle is defined as the ratio of the side which is adjacent to the angle (θ) to the hypotenuse (longest side) in the triangle.

  3. Tangent: Tangent of an angle is defined as the ratio of the side which is opposite to the angle (θ) to the adjacent in the triangle.


You might have heard about SOH, CAH and TOA!


SOH, COH and TOA is a mnemonic way or trick to remember the three basic trigonometric ratios defined by the trigonometric ratio definition.


SOH stands for Sine equals Opposite over Hypotenuse. \[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]

CAH stands for Cosine equals Adjacent over Hypotenuse. \[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

TOA stands for Tangent equals Opposite over Adjacent. \[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]


Here’s a table showing the value of each ratio with respect to different angles . These ratios are used in different calculations and are important for solving various problems.


Table showing the value of each ratio with respect to different angles.


What are Trigonometric Ratios?

Angle

    0o

    30o

    45o

    60o

  90o

Sin \[\theta \]

0

\[\frac{1}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

1

Cos \[\theta \]

1

\[\frac{{\sqrt 3 }}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{1}{2}\]

0

Tan \[\theta \]

0

\[\frac{1}{{\sqrt 3 }}\]

1

\[\sqrt 3 \]

\[\infty \]

Cot \[\theta \]

\[\infty \]

\[\sqrt 3 \]

1

    \[\frac{1}{{\sqrt 3 }}\]

0

Sec \[\theta \]

1

\[\frac{2}{{\sqrt 3 }}\]

\[\sqrt 2 \]

2

\[\infty \]

Cosec \[\theta \]

\[\infty \]

2

\[\sqrt 2 \]

\[\frac{2}{{\sqrt 3 }}\]

1


A Few Applications of Trigonometry

  1. Trigonometry is used in cartography which is the creation of maps.

  2. It has its applications in satellite systems.

  3. It is used in aviation industries.

  4. The functions of trigonometry are used to describe the sound and light waves.


Questions to be Solved on All Trigonometric Ratios

Question 1) In the triangle given below state whether a, b and c are the hypotenuse, opposite or adjacent with respect to the marked angle.


Solution) Here, the side directly opposite to the right angle is the hypotenuse.

So, in the triangle given above, b is the hypotenuse.


The opposite side relative to the marked angle is directly opposite to the marked angle, therefore c is the opposite side.


And the remaining side a is obviously the adjacent side.

  • Adjacent side – a

  • Opposite side – c

  • Hypotenuse – b


Question 2) Find out what is wrong with the triangle given below and explain why.


Angle \[\theta \]= 30o


Solution) We know that the hypotenuse is the longest side of all the three sides and is the side directly opposite to the right angle.


Here, the hypotenuse =4 and the adjacent is 12, which is not possible.


Question 3) Calculate sin(A) from the triangle given below. (image will be uploaded soon)


Solution) We know the formula of \[\sin \left( A \right) = \frac{{opposite}}{{Hypotenuse}}\]

In the given question, 

Opposite = 11

Hypotenuse = 61

Then, sin(A) =\[\frac{{11}}{{61}}\]


Question 4) Calculate the value of sin 15o.


Solution) We have to find the value of sin 15°, which can also be written as sin 

(45o -30o),

Since, \[{\mathbf{sin}}\left( {{\mathbf{A}} - {\mathbf{B}}} \right){\text{ }} = {\mathbf{sinAcosB}}{\text{ }} - {\mathbf{cosAsin}}{\text{ }}{\mathbf{B}},\]

= sin 45°cos 30o - cos 45o  sin 30o

Putting the values of sin 45° cos 30° from the table above,

= \[\frac{1}{{\sqrt 2 }} \times \frac{{\sqrt 3 }}{2} - \frac{1}{{\sqrt 2 }} \times \frac{1}{2} = \frac{{\sqrt {3 - 1} }}{{2\sqrt 2 }}\].

FAQ (Frequently Asked Questions)

1) What are the six maths Trigonometric Ratios?

 There are six maths trigonometric ratios for any triangle namely 

  • Sine(sin)

  • Cosine(cos)

  • Tangent(tan)

  • Cosecant(cosec)

  • Secant(sec)

  • Cotangent(tan)

The Table Below Shows the Formulas for all Trigonometric Ratios-

Name

Abbreviation 

Relationship 

Sine

Sin

Sin (θ) = Opposite/Hypotenuse

Cosine

Cos

Cos (θ) = Adjacent/Hypotenuse 

Tangent

Tan

Tan (θ) = Opposite/Adjacent 

Cosecant

Cosec

Cosec (θ) = Hypotenuse/Opposite 

Secant

Sec

Sec (θ) = Hypotenuse/Adjacent

Cotangent

Cot

Cot (θ) = Adjacent/Opposite

2) What are the three trigonometric ratios?

There are six trigonometric ratios according to the trigonometric ratio definition maths, but these three are the basic ratios


Name

Abbreviation 

Relationship 

Sine

Sin

Sin (θ) = Opposite/Hypotenuse

Cosine

Cos

Cos (θ) = Adjacent/Hypotenuse 

Tangent

Tan

Tan (θ) = Opposite/Adjacent 

3) What is SOH, CAH, TOA mean?

Answer) SOH, CAH and TOH is basically a trick or a way to remember how to compute the trigonometric ratios in mathematics,

  • SOH stands for Sine equals Opposite over Hypotenuse. (Sin (θ)= Opposite/Hypotenuse)

  • CAH stands for Cosine equals Adjacent over Hypotenuse. ( Cos (θ)= Adjacent/Hypotenuse

TOA stands for Tangent equals Opposite over Adjacent. (Tan (θ)= Opposite/Adjacent)

4. What are the primary trigonometric ratios?

Out of all trigonometric ratios according to the trigonometric ratio definition maths, but these three are the basic ratios or the primary ratios 


Name

Abbreviation 

Relationship 

Sine

Sin

Sin (θ) = Opposite/Hypotenuse

Cosine

Cos

Cos (θ) = Adjacent/Hypotenuse 

Tangent

Tan

Tan (θ) = Opposite/Adjacent