Introduction to Trigonometry Class 10 Notes Maths Chapter 8

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Terminologies related to Trigonometry

• Trigonometry is the science of relationships between the sides and angles of a right-angled triangle.
•  Ratios of sides of right triangle are called trigonometric ratios.
• Consider triangle ABC right-angled at B. These ratios are always defined with respect to acute angle ‘A’ or angle ‘C.

• If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of an angle can be easily determined.

Trigonometric Ratios

How to identify sides?Identify the angle with respect to which the t-ratios have to be calculated. Sides are always labelled with respect to the ‘θ’ being considered.

Let us take two cases:

In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six t-Ratios with respect to the sides.

Note from above six relationships:

1. cosecant A = 1/sinA  
2. secant A = 1/cosineA
3. cotangent A = 1/tanA,

However, it is very tedious to write full forms of t-ratios, therefore the abbreviated notations are:

1. sine A is sin A
2. cosine A is cos A
3. tangent A is tan A
4. cosecant A is cosec A
5. secant A is sec A
6. cotangent A is cot A

Example: If in a right-angled triangle ABC, right-angled at B, hypotenuse AC = 5cm, base BC = 3cm and perpendicular AB = 4cm and if ∠ACB = θ, then find tan θ, sin θ and cos θ.

Solution:

Given, 

In ∆ABC, 

Hypotenuse, AC = 5cm 

Base, BC = 3cm 

Perpendicular, AB = 4cm

Then, 

tan θ = Perpendicular/Base = 4/3

Sin θ = Perpendicular/Hypotenuse = AB/AC = ⅘

Cos θ = Base/Hypotenuse = BC/AC = ⅗

Trigonometric Identities

An equation involving trigonometric ratio of angle(s) is called a trigonometric identity, if it is true for all values of the angles involved. These are:

1. sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
2. cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
3. sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1
4. sin θ cosec θ = 1 ⇒ cos θ sec θ = 1 ⇒ tan θ cot θ = 1\

Example: Express the ratios $\backslash cos\; A$cosA, $\backslash tan\; A$tanA, and $\backslash sec\; A$secA in terms of $\backslash sin\; A$sinA. 

Solution: Since $\backslash cos^2\; A\; +\; \backslash sin^2\; A\; =\; 1$cos²A + sin²A = 1, therefore: 

cos²A = 1 − sin²A

i.e., cosA = ±√1−sin²A

This gives:

cosA = √1 − sin²A

Hence, 

and

Note: A trigonometric ratio only depends upon the angle ‘θ’ and stays the same for same angle of different sized right triangles.

Values of Trigonometric Ratios of Specified Angles 

The value of sin θ and cos θ can never exceed 1 (one) as opposite side is 1. Adjacent side can never be greater than hypotenuse since hypotenuse is the longest side in a right-angled ∆.

Example: If tan θ + cot θ = 5, find the value of tan2θ + cotθ.

Solution:

tan θ + cot θ = 5 … [Given

tan²θ + cot²θ + 2 tan θ cot θ = 25 … [Squaring both sides

tan²θ + cot²θ + 2 = 25

∴ tan²θ + cot²θ = 23

Example: If sec 2A = cosec (A – 27°) where 2A is an acute angle, find the measure of ∠A. 

Solution:

sec 2A = cosec (A – 27°)

cosec(90° – 2A) = cosec(A – 27°) …[∵ sec θ = cosec (90° – θ)

90° – 2A = A – 27°

90° + 27° = 2A + A

⇒ 3A = 117°

∴ ∠A = 117°/3 = 39°