Chapter: Trigonometry - Class 10

The term 'trigonometry' originates from the Greek words 'tri' (which means three), 'gon' (which means sides) and 'metron' (which means measure). Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. It explores the properties and functions of angles as well as the connections between angles and the lengths of sides. This chapter includes trigonometric ratios of the angle, trigonometric identities and angles of elevation and depression.

Trigonometric Equation

A trigonometric equation is a mathematical equation that involves one or more trigonometric functions of an unknown angle.

Examples:

Trigonometric Ratios

Trigonometric ratios are mathematical relationships between the angles of a right triangle and the ratios of the lengths of its sides. There are six primary trigonometric ratios.

Reciprocal Identities

The Reciprocal Identities are shown as:

Example: If △ABC is right-angled at B, AB = 6 cm and AC − BC  = 2 cm, then what is the value of sin A − cot C?

a) −8/5
b) −8/15
c) 8/5
d) 8/15

Answer: b) −8/15

Explanation:

We are given that AB = 6 cm

AC − BC = 2 cm

⇒ AC = 2 + BC

In △ABC, using Pythagoras theorem

AC² = AB² + BC²

Substituting the value of AC = 2 + BC in the above equation

⇒ (2 + BC)² = AB² + BC²

⇒ 4 + 4BC + BC² = AB² + BC²

⇒ 4 + 4BC = (6)² + BC² − BC²

⇒ 4 + 4BC = 36

⇒ 4BC = 36 − 4

⇒ 4BC = 32

⇒ BC = 32/4

⇒ BC = 8 cm

Now, AC = 2 + BC

⇒ AC = 2 + 8

⇒ AC = 10 cm

Trigonometric Ratios of Some Specific Angles

The trigonometric ratios of some specific angles is shown in the table below:

Example: What is the value of sin 30° cos 30° + sin 60° cos 45° tan 45°?

a) 
b)
c) 
d) 

Answer: d)

Explanation: We know that:

Important Identities of Trigonometry

Trigonometric Identities

Trigonometric identities are as follows:

→ sin² A + cos² A = 1

→ sec² A − tan² A  = 1

→ cosec² A− cot²A = 1
Note: sin² A = (sin A)²
Similarly, tan² A = (tan A)²

This is applicable to all the trigonometric ratios.

Example: What is the value of the following trigonometric expression?

a) 1/3
b) 1
c) 1/5
d) 1/7

Answer: a) 1/3

Explanation: We know that
sin² A + cos² A = 1
sec² A − tan² A = 1

Trigonometric Sign Functions

Trigonometric sign functions are as follows:

→ sin (−θ) = − sin θ [θ is the angle.]
→ cos (−θ) = cos θ
→ tan (−θ) = − tan θ
→ cosec (−θ) = − cosec θ
→ sec (−θ) = sec θ   
→ cot (−θ) = − cot θ 

Example: What is the value of cosec (− 60°) using the identity cosec (−θ) = − cosec θ?

a) −√3 /2 
b) √3 /2 
c) −2/√3
d) 2/√3

Answer: c) −2/√3

Explanation: Given: cosec (−θ) = − cosec θ 

cosec (− 60°) = − cosec (60°)

We know that cosec 60° = 1/sin 60° = 2/√3

→ cosec (− 60°) = − cosec (60°) = −2/√3  

Periodic Identities

Periodic identities are true when n ∈ z. They are as follows:

→ sin (2nπ + θ) = sin θ
→ cos (2nπ + θ) = cos θ 
→ tan (2nπ + θ) = tan θ 
→ cot (2nπ + θ) = cot θ 
→ sec (2nπ + θ) = sec θ 
→ cosec (2nπ + θ) = cosec θ

Note: π radians = 180°

Example: What is the value of sin (1500°) using the identity sin (2nπ + θ) = sin θ, where n ∈ z?

a) √3/2
b) 1/√2
c) 1/2
d) 2/√3

Answer: a) √3/2

Explanation: We know that sin (2nπ + θ) = sin θ …………….(1)

sin (1500°) = sin (1440° + 60°)  …………….(2)

Comparing (1) and (2),

→ 2nπ = 1440°

→ 2 × n × 180° = 1440° [π radians = 180°]

→ n = 1440°/360°

→ n = 4

→ sin (1500°) = sin (1440° + 60°)
                      = sin (8π + 60°) [2nπ =2 × 4 × π = 8π]
                       = sin 60° [sin (2nπ + θ) = sin θ]
                        = √3 /2

Sum and Difference of Two Angles

Sum and difference of two angles are as follows:

→ sin (A + B) = sin A cos B + cos A sin B
→ sin (A − B) = sin A cos B − cos A sin B
→ cos (A + B) = cos A cos B − sin A sin B
→ cos (A − B) = cos A cos B + sin A sin B
→ tan (A + B) = ^{tan A + tan B}⁄_{1 - A-tan A tan B}
→ tan (A − B) = ^{tan A - tan B}⁄_{1 + A-tan A tan B}

Example: What is the value of sin 15° using the identity sin (A − B) = sin A cos B − cos A sin B?

a) 
b) 
c) 
d) 

Answer: c) 

Explanation:

Double Angle Formulas

Double angle formulae are as follows:

Example: If sin 2x + cos 2x = 1/4, then what is the value of sin x + cos x using the identities sin 2A = 2 sin A cos A and cos 2A = 2 cos² A − 1?

a) 5cos x/4
b) 5sec x/4
c) 5cos x/8
d) 5sec x/8

Answer: d) 5sec x/8

Explanation: We know that

sin 2A = 2 sin A cos A

And

cos 2A = 2 cos² A − 1

We are given that: sin 2x + cos 2x = 1/4

2 sin x cos x + 2 cos² x − 1 = 1/4

2 sin x cos x + 2 cos² x = 1/4 + 1

2 cos x (sin x + cos x) = 5/4

cos x (sin x + cos x) = 5/8

sin x + cos x = 5sec x/8

Triple Angle Formulas

Triple angle formulae are as follows:

Note: sin³ A = (sin A)³
This is applicable to all the trigonometric ratios.   

Example: What is the value of tan 135°?

a) −1 / 2
b) 1 / 2
c) 1
d) −1

Answer: d) −1

Explanation:

Complementary Ratios

Signs of trigonometric functions in the different quadrants are shown as follows:

Complementary ratios in trigonometry involve the relationships between the trigonometric functions of two angles that add up to 90°.

Angles of Elevation and Depression

Angle of Elevation: The angle of elevation is the angle formed between the line of sight (typically from a point below) and the horizontal line or plane. It is the angle at which an observer must look upward from the horizontal to see a point or object that is higher than their level.

Angle of Depression: The angle of depression is the angle formed between a downward line of sight from a point of observation and the horizontal plane or line. It is the angle at which an observer must look downward to see an object below the horizontal level.

The angles of elevation and depression are key concepts in trigonometry that find applications in various real-world scenarios involving the calculation of heights and distances in a wide range of practical applications.