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Introduction to Trigonometric Ratios of a Triangle

Trigonometry is all about triangles or to be more precise about the relation between the angles and sides of a right-angled triangle. There are three sides of a triangles named as Hypotenuse, Adjacent, and Opposite. The ratio between these sides based on the angle between them are called Trigonometric Ratios.
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10

As given in the figure in a right angle triangle

  • The side opposite to the right angle is called the hypotenuse
  • The side opposite to an angle is called the opposite side
    (i) For angle C opposite side is AB
    (ii) For angle A opposite side is BC
  • The side adjacent to an angle is called the adjacent side
    (i) For angle C adjacent side is BC
    (ii) For angle A adjacent side is AB

Trigonometric ratios

There are 6 basic trigonometric relations that form the basics of trigonometry. These 6 trigonometric relations are ratios of all the different possible combinations in a right-angled triangle.
These trigonometric ratios are called

  • Sine
  • Cosine
  • Tangent
  • Cosecant
  • Secant
  • Cotangent

The mathematical symbol θ is used to denote the angle.
A. Sine (sin)
Sine of an angle is defined by the ratio of lengths of sides which is opposite to the angle and the hypotenuse. It is represented as sinθ
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10

B. Cosine (cos) 
Cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. It is represented as cosθ
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10C. Tangent (tan)
Tangent of an angle is defined by the ratio of length of sides which is opposite to the angle and the side which is adjacent to the angle. It is represented as tanθ
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10D. Cosecant (csc) 
Cosecent of an angle is defined by the ratio of length of the hypotenuse and the side opposite the angle. It is represented as cscθ
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10E. Secant(sec) 
Secant of an angle is defined by the ratio of length of the hypotenuse and the side and the side adjacent to the angle. It is represented as secθ
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10F. Cotangent(cot) 
Cotangent  of an angle is defined by the ratio of length of sides which is adjacent to the angle and the side which is opposite to the angle. It is represented as cotθ.
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10

Trigonometric table

Trigonometric Ratio

Abbreviation

Formula

sine

sin

Opposite/Hypotenuse

cosine

cos

Adjacent/Hypotenuse

tangent

tan

Opposite/Adjacent

cosecant

csc

Hypotenuse/Opposite

secant

sec

Hypotenuse/Adjacent

cotangent

cot

Adjacent/Opposite

Solving for a side in right triangles with trigonometry
This is one of the most basic and useful use of trigonometry using the trigonometric ratios mentioned is to find the length of a side of a right-angled triangle But to do, so we must already know the length of the other two sides or an angle and length of one side.

Steps to follow if one side and one angle are known:

  • Choose a trigonometric ratio which contains the given side and the unknown side
  • Use algebra to find the unknown side

Example: In a right angled ΔABC ∠B — 30 length of side A B is 4 find length of BC. given tan30 = 1/√3
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10Solution:
Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10

Steps to follow if two sides are known:

  • Mark the known sides as adjacent, opposite or hypotenuse with respective to anyone of the acute angles in the triangle.
  • Decide on which trigonometric ratio can be found out from the above table.
  • Find the angle (X)
  • Use an trigonometric ratio with respect to X which is a ratio of a known side and an unknown side.
  • Use algebra to find the unknown side.
The document Introduction: Trigonometric Ratios | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Introduction: Trigonometric Ratios - Mathematics (Maths) Class 10

1. What are trigonometric ratios of a triangle?
Ans. Trigonometric ratios of a triangle are mathematical functions that relate the angles of a triangle to the lengths of its sides. The three main trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
2. How do you calculate the sine of an angle in a triangle?
Ans. To calculate the sine of an angle in a triangle, divide the length of the side opposite to the angle by the length of the hypotenuse. For example, if the opposite side is 'a' and the hypotenuse is 'c', then the sine of the angle is given by sin(angle) = a/c.
3. What is the cosine ratio in a triangle?
Ans. The cosine ratio in a triangle is a trigonometric function that relates the length of the adjacent side to the length of the hypotenuse. It is calculated by dividing the length of the adjacent side by the length of the hypotenuse. Mathematically, cos(angle) = adjacent side/hypotenuse.
4. How do you find the tangent ratio in a triangle?
Ans. To find the tangent ratio in a triangle, divide the length of the side opposite to the angle by the length of the adjacent side. In other words, if the opposite side is 'a' and the adjacent side is 'b', then the tangent of the angle is given by tan(angle) = a/b.
5. Can trigonometric ratios be used in any triangle?
Ans. No, trigonometric ratios can only be used in right-angled triangles. The ratios are defined based on the sides of a right-angled triangle and the relationship between the angles and those sides. In other types of triangles, different trigonometric functions need to be used, such as the law of sines or the law of cosines.
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