Introduction To Trigonometry Class 10

Calculus is an important branch of Mathematics used across a wide range of disciplines like Engineering, Economics, Physics and more. Even though the complexities of calculus are introduced much late, the basic process of understanding its concepts begins in class 10. Introduction to Trigonometry class 10 briefs students with some simple concepts of angles, identities and ratios. These trigonometric NCERT solutions can help students to prepare for their board exams and assist them with the fundamental understanding of this topic. Let’s begin! 

Introduction to Trigonometry Class 10 Overview

Trigonometry word has been derived from the Greek word combining: ‘tri’ implies three, ‘gon’ as a side and ‘metron’ as a measurement.

Trigonometry is the branch of Mathematics that studies the techniques to find distance and heights, sides, and angles within a triangle using special techniques. In earlier times, astronomers used these formulas of trigonometry and other mathematical formulas for the estimated distance of the earth to celestial bodies of sun, moon, and stars. Several Physical sciences and Engineering concepts of this modern age use trigonometry to refine complex problems. 

Students must be aware of the formation of a triangle and its basic properties. Starting this, in the chapter Introduction to Trigonometry class 10, we have a simple example of surroundings that has a right triangle. In the below picture, we can see a boy standing on the second floor of the building and watching the temple on the other side of the river. Now the trigonometry class 10 formulas can precisely help students to find the height and distance in these scenarios where one angle is 90⁰ or right-angled. 

What are Trigonometric Ratios?  

Here is a simple triangle describing the concepts of ratios mentioned in Introduction to Trigonometry class 10. So as per the triangle properties, angle ∠A here represents the acute angle. Now the side opposite to that angle is termed as ‘the side opposite to Angle A’.Here AC represents the hypotenuse of the right triangle presentation. And the side AB is termed as the side adjacent to the original angle A. 

Similarly, if one considers angle C as the acute angle in the triangle, then the positions change.

So taking the example of Angle A in the right angle ABC, trigonometry class 10 formulas are defined as follows:

In NCERT solutions for class 10 maths, these trigonometric ratios are known as sin A, cos A, tan A, cosec A, sec A, and cot A. Also, you must have noticed that the three ratios of cosec A, sec A, cot A, are reciprocals or reverse ratios of the first three ratios with sin A, cos A, and tan A. 

Linear Equations in Two Variables Class 10

Trigonometric Ratios of Some Specific Angles 

With geometry, you must already know about constructions of various trigonometric angles of 0⁰, 30⁰, 45⁰, 60⁰, and 90⁰. Now in this section of Introduction to Trigonometry class 10,  we will define the value of trigonometric values of these angles. Here are the fundamental trigonometry class 10 formulas used in solving height and distance problems. 

θ	0°	30°	45°	60°	90°	180°
Sin	0	1/2	1/2	3/2	1	0
Cos	1	3/2	1/2	1/2	0	-1
Tan	0	1/3	1	3	∞	0
Cot	∞	3	1	1/3	0	∞
Sec	1	2/3	2	2	∞	-1
Cosec	∞	2	2	2/3	1	∞

Trigonometric Ratios of Complementary Angles 

The next topic in Introduction to Trigonometry class 10 Maths is complementary angles. If two angles sum equals 90⁰, then they are known as complementary angles. So in a given triangle ABC, we can see that it is right-angled at B. (grossmancapraroplasticsurgery.com) Hence ∠A and ∠C are complementary angles with ∠A + ∠C = 90°^.

Now, these complementary angles (∠ A + ∠ C = 90° ) form several pairs in a triangle. Similarly, we can also write trigonometric ratios for ∠ C = 90° – ∠ A. Have a look at the table given below for a better understanding:

sin A = BC/AC cos A= AB/AC tan A= BC/AB cosec A = AC/BC sec A= AC/AB cot A = AB/BC

sin (90°–A) = AB/AC cos(90°–A)= BC/AC tan(90°–A)= AB/BC cosec (90°–A) = AC/AB sec (90°–A) = AC/BC cot(90°–A) = BC/AB

Now, if we compare the above two highlighted tables. Equations become:  
sin (90° – A) = AB/AC = cos A
cos (90°–A) = BC/AC = sin A 
tan (90° – A) = AB/BC = cot A
cot (90° – A) = BC/AB = tan A 
sec (90°–A) = AC/BC = cosec A
cosec (90°–A) = AC/AB = sec A 

Hence, these all hold true:
sin (90°– A) = cos A
tan (90° – A) = cot A
sec (90° – A) = cosec A
cos(90°– A) = sin A
cot (90° – A) = tan A
cosec (90° – A) = sec A

Important Trigonometric Identities

The next section of Introduction to Trigonometry class 10 is  Trigonometric identities. These are the specific ratios and equations that involve trigonometric ratios of an angle. Trigonometric identities must hold true for all the given values of angle. For instance, cos² A+sin² A=1 is true, for all A angles 0° ≤ A ≤ 90°. Now if we divide the above equation by AB² then we get the following, 

You can check the above statement holds true for all angles 0° ≤ A < 90°.

Now, we are going to divide the equation by BC², and we see the following changes;

This trigonometric identity also holds true for all angles 0° ≤ A < 90°. Now, these three identities help us to resolve several complex trigonometric issues. 

cos²  A+ sin² A = 1 
1+ tan² A = sec²  A 
cot²A + 1 = cosec² A

There are several formulas of trigonometry that sometimes can create confusion in a student’s mind. But these NCERT solutions for class 10 maths on trigonometric identities can help them understand the concepts and solve complex issues precisely. Introduction to Trigonometry class 10 is an essential part of the class 10 maths syllabus and offers comprehensive insights for solving complex real-life problems. Trigonometry can help students understand the basic concepts of finding height, distance, and interdependence. Students must practice these formulas of trigonometry regularly to assist them in their practical problems. 

Introduction to Trigonometry Class 10 NCERT PDF

Introduction to Trigonometry Class 10 PPT 

Trigonometry Presentation For Class 10 Students from Abhishek Yadav

Introduction to Trigonometry Important MCQ Questions 

1. sin 2B = 2 sin B is true when B is equal to

A. 90°

B. 60°

C. 30°

D. 0°

2. If x and y are complementary angles, then

A. sin x = sin y

B. tan x = tan y

C. cos x = cos y

D. sec x = cosec y

3.  If A and (2A – 45°) are acute angles such that sin A = cos (2A – 45°), then tan A is equal to

A. 0

B. 1/√3

C. 1

D. √3

4.  If sin θ + sin² θ = 1, then cos² θ + cos4 θ = 

A.  -1

B. 0

C. 1

D. 2

5. 5 tan² A – 5 sec² A + 1 is equal to

A. 6

B. -5

C. 1

D. -4

6. If y sin 45° cos 45° = tan2 45° – cos2 30°, then y = …

A. -½

B. ½

C. -2

D. 2

7. The value of cos 0°. cos 1°. cos 2°. cos 3°… cos 89° cos 90° is

A. 1

B. -1

C. 0

D. 1/√2 

8. What is the maximum value of 1/cosec A?

A. 0

B. 1

C. ½ 

D. 2 

9. What is the minimum value of sin A, 0 ≤ A ≤ 90°

A. -1

B. 0

C. 1 

D. ½

10.  If cos 9A = sin A and 9A < 90°, then the value of tan 5A is

A. 0

B. ½ 

C. 1/√2 

D. 1

Answers 

1. D
2. D
3. C
4. C
5. D
6. B
7. C
8. B 
9. B
10. D 

Introduction to Trigonometry Class 10 Assignment 

1. If tan θ + cot θ = 5, find the value of tan2θ + cotθ. (2012)
2. If sec 2A = cosec (A – 27°) where 2A is an acute angle, find the measure of ∠A.
3. If sin θ – cos θ = 0, find the value of sin4 θ + cos4 θ.
4. If sec θ + tan θ = 7, then evaluate sec θ – tan θ
5. Find the value of:
6. If θ = 45°, then what is the value of 2 sec²θ + 3 cosec²θ?
7. Evaluate: sin² 19° + sin²71°
8. What happens to the value of cos when increases from 0° to 90°?
9. If in a right-angled ∆ABC, tan B = 12/5, then find sin B.
10. If ∆ABC is right-angled at B, what is the value of sin (A + C).
11. Evaluate tan 15° . tan 25° , tan 60° . tan 65° . tan 75° – tan 30°
12. Express cot 75° + cosec 75° in terms of trigonometric ratios of angles between 0° and 30°
13. If x = p sec θ + q tan θ and y = p tan θ + q sec θ, then prove that x² – y² = p² – q² 
14. Given 2 cos 3θ = √3, find the value of θ.
15. If sec θ + tan θ = p, prove that sin θ = (p² – 1)/(p² + 1).

This guide on Introduction to Trigonometry class 10 Maths offers a step by step learning curve to score maximum marks in exams. If you have any queries around careers, courses or stream selection, contact Leverage Edu for expert guidance.

Calculus is an important branch of Mathematics used across a wide range of disciplines like Engineering, Economics, Physics and more. Even though the complexities of calculus are introduced much late, the basic process of understanding its concepts begins in class 10. Introduction to Trigonometry class 10 briefs students with some simple concepts of angles, identities and ratios. These trigonometric NCERT solutions can help students to prepare for their board exams and assist them with the fundamental understanding of this topic. Let’s begin! 

Introduction to Trigonometry Class 10 Overview

Trigonometry word has been derived from the Greek word combining: ‘tri’ implies three, ‘gon’ as a side and ‘metron’ as a measurement.

Trigonometry is the branch of Mathematics that studies the techniques to find distance and heights, sides, and angles within a triangle using special techniques. In earlier times, astronomers used these formulas of trigonometry and other mathematical formulas for the estimated distance of the earth to celestial bodies of sun, moon, and stars. Several Physical sciences and Engineering concepts of this modern age use trigonometry to refine complex problems. 

Students must be aware of the formation of a triangle and its basic properties. Starting this, in the chapter Introduction to Trigonometry class 10, we have a simple example of surroundings that has a right triangle. In the below picture, we can see a boy standing on the second floor of the building and watching the temple on the other side of the river. Now the trigonometry class 10 formulas can precisely help students to find the height and distance in these scenarios where one angle is 90⁰ or right-angled. 

What are Trigonometric Ratios?  

Here is a simple triangle describing the concepts of ratios mentioned in Introduction to Trigonometry class 10. So as per the triangle properties, angle ∠A here represents the acute angle. Now the side opposite to that angle is termed as ‘the side opposite to Angle A’.Here AC represents the hypotenuse of the right triangle presentation. And the side AB is termed as the side adjacent to the original angle A. 

Similarly, if one considers angle C as the acute angle in the triangle, then the positions change.

So taking the example of Angle A in the right angle ABC, trigonometry class 10 formulas are defined as follows:

In NCERT solutions for class 10 maths, these trigonometric ratios are known as sin A, cos A, tan A, cosec A, sec A, and cot A. Also, you must have noticed that the three ratios of cosec A, sec A, cot A, are reciprocals or reverse ratios of the first three ratios with sin A, cos A, and tan A. 

Linear Equations in Two Variables Class 10

Trigonometric Ratios of Some Specific Angles 

With geometry, you must already know about constructions of various trigonometric angles of 0⁰, 30⁰, 45⁰, 60⁰, and 90⁰. Now in this section of Introduction to Trigonometry class 10,  we will define the value of trigonometric values of these angles. Here are the fundamental trigonometry class 10 formulas used in solving height and distance problems. 

θ	0°	30°	45°	60°	90°	180°
Sin	0	1/2	1/2	3/2	1	0
Cos	1	3/2	1/2	1/2	0	-1
Tan	0	1/3	1	3	∞	0
Cot	∞	3	1	1/3	0	∞
Sec	1	2/3	2	2	∞	-1
Cosec	∞	2	2	2/3	1	∞

Trigonometric Ratios of Complementary Angles 

The next topic in Introduction to Trigonometry class 10 Maths is complementary angles. If two angles sum equals 90⁰, then they are known as complementary angles. So in a given triangle ABC, we can see that it is right-angled at B. (grossmancapraroplasticsurgery.com) Hence ∠A and ∠C are complementary angles with ∠A + ∠C = 90°^.

Now, these complementary angles (∠ A + ∠ C = 90° ) form several pairs in a triangle. Similarly, we can also write trigonometric ratios for ∠ C = 90° – ∠ A. Have a look at the table given below for a better understanding:

sin A = BC/AC cos A= AB/AC tan A= BC/AB cosec A = AC/BC sec A= AC/AB cot A = AB/BC

sin (90°–A) = AB/AC cos(90°–A)= BC/AC tan(90°–A)= AB/BC cosec (90°–A) = AC/AB sec (90°–A) = AC/BC cot(90°–A) = BC/AB

Now, if we compare the above two highlighted tables. Equations become:  
sin (90° – A) = AB/AC = cos A
cos (90°–A) = BC/AC = sin A 
tan (90° – A) = AB/BC = cot A
cot (90° – A) = BC/AB = tan A 
sec (90°–A) = AC/BC = cosec A
cosec (90°–A) = AC/AB = sec A 

Hence, these all hold true:
sin (90°– A) = cos A
tan (90° – A) = cot A
sec (90° – A) = cosec A
cos(90°– A) = sin A
cot (90° – A) = tan A
cosec (90° – A) = sec A

Important Trigonometric Identities

The next section of Introduction to Trigonometry class 10 is  Trigonometric identities. These are the specific ratios and equations that involve trigonometric ratios of an angle. Trigonometric identities must hold true for all the given values of angle. For instance, cos² A+sin² A=1 is true, for all A angles 0° ≤ A ≤ 90°. Now if we divide the above equation by AB² then we get the following, 

You can check the above statement holds true for all angles 0° ≤ A < 90°.

Now, we are going to divide the equation by BC², and we see the following changes;

This trigonometric identity also holds true for all angles 0° ≤ A < 90°. Now, these three identities help us to resolve several complex trigonometric issues. 

cos²  A+ sin² A = 1 
1+ tan² A = sec²  A 
cot²A + 1 = cosec² A

There are several formulas of trigonometry that sometimes can create confusion in a student’s mind. But these NCERT solutions for class 10 maths on trigonometric identities can help them understand the concepts and solve complex issues precisely. Introduction to Trigonometry class 10 is an essential part of the class 10 maths syllabus and offers comprehensive insights for solving complex real-life problems. Trigonometry can help students understand the basic concepts of finding height, distance, and interdependence. Students must practice these formulas of trigonometry regularly to assist them in their practical problems. 

Introduction to Trigonometry Class 10 NCERT PDF

Introduction to Trigonometry Class 10 PPT 

Trigonometry Presentation For Class 10 Students from Abhishek Yadav

Introduction to Trigonometry Important MCQ Questions 

1. sin 2B = 2 sin B is true when B is equal to

A. 90°

B. 60°

C. 30°

D. 0°

2. If x and y are complementary angles, then

A. sin x = sin y

B. tan x = tan y

C. cos x = cos y

D. sec x = cosec y

3.  If A and (2A – 45°) are acute angles such that sin A = cos (2A – 45°), then tan A is equal to

A. 0

B. 1/√3

C. 1

D. √3

4.  If sin θ + sin² θ = 1, then cos² θ + cos4 θ = 

A.  -1

B. 0

C. 1

D. 2

5. 5 tan² A – 5 sec² A + 1 is equal to

A. 6

B. -5

C. 1

D. -4

6. If y sin 45° cos 45° = tan2 45° – cos2 30°, then y = …

A. -½

B. ½

C. -2

D. 2

7. The value of cos 0°. cos 1°. cos 2°. cos 3°… cos 89° cos 90° is

A. 1

B. -1

C. 0

D. 1/√2 

8. What is the maximum value of 1/cosec A?

A. 0

B. 1

C. ½ 

D. 2 

9. What is the minimum value of sin A, 0 ≤ A ≤ 90°

A. -1

B. 0

C. 1 

D. ½

10.  If cos 9A = sin A and 9A < 90°, then the value of tan 5A is

A. 0

B. ½ 

C. 1/√2 

D. 1

Answers 

1. D
2. D
3. C
4. C
5. D
6. B
7. C
8. B 
9. B
10. D 

Introduction to Trigonometry Class 10 Assignment 

1. If tan θ + cot θ = 5, find the value of tan2θ + cotθ. (2012)
2. If sec 2A = cosec (A – 27°) where 2A is an acute angle, find the measure of ∠A.
3. If sin θ – cos θ = 0, find the value of sin4 θ + cos4 θ.
4. If sec θ + tan θ = 7, then evaluate sec θ – tan θ
5. Find the value of:
6. If θ = 45°, then what is the value of 2 sec²θ + 3 cosec²θ?
7. Evaluate: sin² 19° + sin²71°
8. What happens to the value of cos when increases from 0° to 90°?
9. If in a right-angled ∆ABC, tan B = 12/5, then find sin B.
10. If ∆ABC is right-angled at B, what is the value of sin (A + C).
11. Evaluate tan 15° . tan 25° , tan 60° . tan 65° . tan 75° – tan 30°
12. Express cot 75° + cosec 75° in terms of trigonometric ratios of angles between 0° and 30°
13. If x = p sec θ + q tan θ and y = p tan θ + q sec θ, then prove that x² – y² = p² – q² 
14. Given 2 cos 3θ = √3, find the value of θ.
15. If sec θ + tan θ = p, prove that sin θ = (p² – 1)/(p² + 1).

This guide on Introduction to Trigonometry class 10 Maths offers a step by step learning curve to score maximum marks in exams. If you have any queries around careers, courses or stream selection, contact Leverage Edu for expert guidance.