CBSE Class 10 Maths – Introduction to Trigonometry Notes (NCERT Solutions + Important Questions)

CBSE Class 10 Maths Notes – Chapter 8 Introduction to Trigonometry

CBSE Class 10 Notes for Mathematics Chapter 8 introduce students to trigonometry—the branch of mathematics dealing with relationships between sides and angles of triangles. These CBSE Class 10 Maths Notes are 100% NCERT aligned and essential for board exam preparation.

1. Introduction to Trigonometry

The word "trigonometry" comes from Greek words tri (three), gon (sides), and metron (measure). In CBSE Class 10 Maths Notes, we focus on right-angled triangles where one angle is exactly 90°.

Why This Chapter Matters:

• Weightage: 4-6 marks in CBSE Board Exam
• NCERT Based: 100% aligned with textbook
• Foundation: Essential for Class 11-12 and competitive exams
• Question Types: MCQs, Short Answer, Long Answer, Case-Study

2. Chapter Overview

CBSE Class 10 Notes for Introduction to Trigonometry cover:

• Trigonometric Ratios – sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ
• Standard Values – For angles 0°, 30°, 45°, 60°, 90°
• Trigonometric Identities – Three fundamental identities
• Complementary Angles – Relationships between (90° – θ) and θ
• Identity Proofs – High-weightage board exam problems

3. Key Concepts (NCERT Aligned)

3.1 Parts of a Right-Angled Triangle

Term	Definition	Hindi
Hypotenuse	Longest side opposite to right angle	कर्ण
Perpendicular	Side opposite to angle θ	लंब
Base	Side adjacent to angle θ	आधार

3.2 Trigonometric Ratios Class 10

Primary Ratios

sin θ = PerpendicularHypotenuse = PH
cos θ = BaseHypotenuse = BH
tan θ = PerpendicularBase = PB

Reciprocal Ratios

cosec θ = 1sin θ = HP
sec θ = 1cos θ = HB
cot θ = 1tan θ = BP

Memory Trick: "Pandit Badri Prasad Har Har Bole"

• Pandit (Perpendicular) - Har (Hypotenuse) = sin
• Badri (Base) - Har (Hypotenuse) = cos
• Prasad (Perpendicular) - Bole (Base) = tan

3.3 Trigonometric Identities Class 10

Fundamental Identities (Must Memorize)

Identity 1: sin²θ + cos²θ = 1
Identity 2: 1 + tan²θ = sec²θ
Identity 3: 1 + cot²θ = cosec²θ

Reciprocal Relations

1. sin θ × cosec θ = 1      2. sin θ = 1cosec θ      3. cosec θ = 1sin θ
4. cos θ × sec θ = 1      5. cos θ = 1sec θ      6. sec θ = 1cos θ
7. tan θ × cot θ = 1      8. tan θ = 1cot θ      9. cot θ = 1tan θ
10. tan θ = sin θcos θ      11. cot θ = cos θsin θ

erived Identities from Fundamental

12. sin²θ + cos²θ = 1
13. sin²θ = 1 − cos²θ
14. cos²θ = 1 − sin²θ
15. sec²θ − tan²θ = 1
16. tan²θ = sec²θ − 1
17. sec²θ = 1 + tan²θ
18. cosec²θ − cot²θ = 1
19. cot²θ = cosec²θ − 1
20. cosec²θ = 1 + cot²θ

Proof of Identity 1:

Using Pythagoras theorem: P² + B² = H²

Divide by H²: P²H² + B²H² = 1

Therefore: sin²θ + cos²θ = 1 Proved.

3.4 Standard Trigonometric Values Table

θ	0°	30°	45°	60°	90°
sin θ	0	12	1√2	√32	1
cos θ	1	√32	1√2	12	0
tan θ	0	1√3	1	√3	Not defined
cosec θ	Not defined	2	√2	2√3	1
sec θ	1	2√3	√2	2	Not defined
cot θ	Not defined	√3	1	1√3	0

Ratios	When θ is increase then value is
sin θ, tan θ, sec θ	Increase
cos θ, cot θ, cosec θ	Decrease
π Radian = 180°

3.5 Complementary Angle Relations

Two angles are complementary if their sum equals 90°.

Complementary Formulas

sin(90° − θ) = cos θ     cosec(90° − θ) = sec θ
cos(90° − θ) = sin θ     sec(90° − θ) = cosec θ
tan(90° − θ) = cot θ     cot(90° − θ) = tan θ

4. Important Formulas Box

Trigonometric Ratios

sin θ = PH     cosec θ = HP
cos θ = BH     sec θ = HB
tan θ = PB     cot θ = BP

When to Use Identities

• sin²θ + cos²θ = 1 – When converting sin to cos or vice versa
• 1 + tan²θ = sec²θ – When tan or sec is involved
• 1 + cot²θ = cosec²θ – When cot or cosec appears

5. Solved Examples (CBSE Pattern)

Example 1: Finding Ratio (Easy)

Question: In ΔABC, right-angled at B, AB = 24 cm, BC = 7 cm. Find sin A and cos A.

Solution:

Using Pythagoras: AC² = AB² + BC² = 24² + 7² = 576 + 49 = 625

AC = 25 cm

For angle A: Perpendicular = BC = 7 cm, Base = AB = 24 cm, Hypotenuse = 25 cm

sin A = 725

cos A = 2425

Example 2: Standard Values (Moderate)

Question: Evaluate: sin 30° cos 60° + cos 30° sin 60°

Solution:

= (12)(12) + (√32)(√32)

= 14 + 34 = 44 = 1

Example 3: Identity Proof (4 Marks)

Question: Prove that: (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ

Solution:

LHS: (sin θ + cosec θ)² + (cos θ + sec θ)²

Expanding: sin²θ + 2 + cosec²θ + cos²θ + 2 + sec²θ

= (sin²θ + cos²θ) + 4 + (1 + cot²θ) + (1 + tan²θ)

= 1 + 4 + 1 + cot²θ + 1 + tan²θ

= 7 + tan²θ + cot²θ = RHS

Hence Proved. ⭐

6. Smart Tricks & Memory Techniques

Finger Trick for Standard Values

Hold left hand, palm facing you:

• Little finger (0°): sin = √02 = 0
• Ring finger (30°): sin = √12 = 12
• Middle finger (45°): sin = √22 = 1√2
• Index finger (60°): sin = √32
• Thumb (90°): sin = √42 = 1

Cos values reverse this order!

identity Proof Strategy

LHS → Simplify using identities → RHS

• Start from complex side
• Convert everything to sin/cos if stuck
• Look for sin²θ + cos²θ = 1 opportunities

7. Most Important Board Questions

★ 1 Mark Questions:

1. If sin θ = 35, find cos θ.
2. Evaluate: tan 45° + cot 45°
3. If sec θ = 54, find tan θ.

★★ 2-3 Mark Questions:

1. If tan θ = 43, evaluate: 3sin θ + 2cos θ3sin θ − 2cos θ
2. Prove that: (cosec θ − cot θ)² = 1 − cos θ1 + cos θ

★★★ 4 Mark Identity Proofs (Highly Important):

1. Prove that: (1 + cot θ − cosec θ)(1 + tan θ + sec θ) = 2
2. Prove that: sec²θ + cosec²θ = sec²θ · cosec²θ
3. Prove that: sin θ − 2sin³θ2cos³θ − cos θ = tan θ

8. Common Mistakes to Avoid

• Confusing Base and Perpendicular: Base is adjacent to angle, Perpendicular is opposite
• Wrong Identity: Remember sin²θ + cos²θ = 1 (not 0)
• Undefined Values: tan 90°, sec 90°, cot 0°, cosec 0° are NOT DEFINED
• Sign Errors: sin(90°−θ) = cos θ (not −cos θ)

9. Practice Section

MCQ: If sin A = 12, then cot A is:
(a) √3   (b) 1√3   (c) √32   (d) 1

HOTS Question: If sin(A+B) = √32 and cos(A−B) = √32, where 0° < A+B ≤ 90° and A > B, find A and B.

10. Frequently Asked Questions

Q: Is Introduction to Trigonometry important for CBSE Class 10 board?
Ans: Yes, it carries 4-6 marks and is essential for advanced mathematics.

Q: How to memorize trigonometric values easily?
Ans: Use the finger trick method and practice the standard values table daily.

Q: Are these CBSE Class 10 Maths Notes based on NCERT?
Ans: Yes, 100% NCERT aligned with CBSE board exam patterns.

Next Chapter: Continue with "Some Applications of Trigonometry – CBSE Class 10 Notes" for Heights and Distances.