Class 10 Mathematics Notes Chapter 8 (Introduction to Trigonometry) – Mathematics Book

Alright students, let's focus on Chapter 8, 'Introduction to Trigonometry'. This is a foundational chapter, not just for your Class 10 exams but also for many competitive government exams where quantitative aptitude is tested. Pay close attention to the concepts and formulas.

Chapter 8: Introduction to Trigonometry - Detailed Notes

1. What is Trigonometry?

• Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles.
• The word 'Trigonometry' is derived from Greek words: 'Tri' (meaning three), 'gon' (meaning sides), and 'metron' (meaning measure).
• In this chapter, we will focus specifically on right-angled triangles.

2. Trigonometric Ratios

• Consider a right-angled triangle ABC, right-angled at B.

• Let's consider an acute angle, say ∠A.

  • The side opposite to ∠A is BC (Perpendicular, P).
  • The side adjacent to ∠A is AB (Base, B).
  • The side opposite to the right angle (∠B) is AC (Hypotenuse, H). Note: Hypotenuse is always the longest side.

• There are six trigonometric ratios defined for the acute angle A:

  • Sine A (sin A) = Side opposite to angle A / Hypotenuse = P / H = BC / AC
  • Cosine A (cos A) = Side adjacent to angle A / Hypotenuse = B / H = AB / AC
  • Tangent A (tan A) = Side opposite to angle A / Side adjacent to angle A = P / B = BC / AB

• The other three ratios are reciprocals of the first three:

  • Cosecant A (cosec A) = 1 / sin A = Hypotenuse / Side opposite to angle A = H / P = AC / BC
  • Secant A (sec A) = 1 / cos A = Hypotenuse / Side adjacent to angle A = H / B = AC / AB
  • Cotangent A (cot A) = 1 / tan A = Side adjacent to angle A / Side opposite to angle A = B / P = AB / BC

• Important Relationships:

  • tan A = sin A / cos A
  • cot A = cos A / sin A

• Mnemonic: To remember the basic ratios (sin, cos, tan): "Pandit Badri Prasad / Har Har Bole" (P/H, B/H, P/B) or "Some People Have / Curly Brown Hair / Turned Permanently Black" (S=P/H, C=B/H, T=P/B).

• Note: The value of trigonometric ratios depends only on the angle, not the size of the triangle. If the angle remains the same, the ratio of sides remains the same even if the lengths of the sides change proportionally.

3. Trigonometric Ratios of Some Specific Angles

• The values of trigonometric ratios for angles 0°, 30°, 45°, 60°, and 90° are crucial and should be memorized.

• Key Observations from the Table:
  • As angle A increases from 0° to 90°, sin A increases from 0 to 1.
  • As angle A increases from 0° to 90°, cos A decreases from 1 to 0.
  • Values of sin A and cos A are always between 0 and 1 (inclusive).
  • tan A can take any positive value (and becomes undefined at 90°).

4. Trigonometric Ratios of Complementary Angles

• Two angles are complementary if their sum is 90°. In a right-angled triangle ABC (right-angled at B), angles A and C are complementary (A + C = 90°).

• The following relationships hold true:

  • sin (90° – A) = cos A
  • cos (90° – A) = sin A
  • tan (90° – A) = cot A
  • cot (90° – A) = tan A
  • sec (90° – A) = cosec A
  • cosec (90° – A) = sec A

• These are useful for simplifying expressions like tan 26° / cot 64° (Since 64° = 90° - 26°, cot 64° = cot(90°-26°) = tan 26°. So the expression equals 1).

5. Trigonometric Identities

• An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle(s) for which the ratios are defined.

• Fundamental Identities: (These are derived using the Pythagoras Theorem: P² + B² = H²)

  1. sin² A + cos² A = 1

     • This implies: sin² A = 1 - cos² A and cos² A = 1 - sin² A

  2. 1 + tan² A = sec² A (for 0° ≤ A < 90°)

     • This implies: sec² A - tan² A = 1 and tan² A = sec² A - 1

  3. 1 + cot² A = cosec² A (for 0° < A ≤ 90°)

     • This implies: cosec² A - cot² A = 1 and cot² A = cosec² A - 1

• These identities are extremely important for proving other trigonometric relations and simplifying complex trigonometric expressions.

Tips for Government Exam Preparation:

1. Memorize: Formulas (ratios, specific angles, complementary angles, identities) are non-negotiable. Create flashcards if needed.
2. Understand Derivations: Knowing how formulas (especially identities) are derived helps in recalling them and tackling tricky problems.
3. Practice: Solve a variety of problems – finding ratios, evaluating expressions, proving identities, problems involving complementary angles. NCERT and NCERT Exemplar are good starting points. Look at previous year questions from relevant exams.
4. Speed & Accuracy: Competitive exams require both. Practice timed quizzes. Learn to apply identities quickly to simplify expressions.
5. Pythagoras Theorem: It's the backbone of deriving ratios and identities. Be comfortable using it.

Multiple Choice Questions (MCQs)

1. In a right-angled triangle ABC, right-angled at B, if sin A = 3/5, what is cos A?
   (A) 4/5
   (B) 3/4
   (C) 5/4
   (D) 5/3

2. The value of tan 60° / cot 30° is:
   (A) 0
   (B) 1
   (C) 2
   (D) √3

3. The value of sin² 45° + cos² 45° is:
   (A) 1/2
   (B) 1
   (C) √2
   (D) 0

4. If sin θ = cos θ, for an acute angle θ, then θ equals:
   (A) 0°
   (B) 30°
   (C) 45°
   (D) 60°

5. The value of tan 10° tan 80° is:
   (A) 0
   (B) 1
   (C) √3
   (D) Undefined

6. 9 sec² A - 9 tan² A is equal to:
   (A) 1
   (B) 9
   (C) 8
   (D) 0

7. If cos A = 1/2, then the value of 2 sec A is:
   (A) 1
   (B) 2
   (C) 4
   (D) 1/4

8. The value of sin 30° + cos 60° - tan 45° is:
   (A) 0
   (B) 1
   (C) 2
   (D) -1/2

9. Which of the following is NOT a trigonometric identity?
   (A) sin² A + cos² A = 1
   (B) 1 + tan² A = sec² A
   (C) 1 + sec² A = tan² A
   (D) cosec² A - cot² A = 1

10. If tan A = 4/3, then cosec A is:
    (A) 3/5
    (B) 4/5
    (C) 5/4
    (D) 5/3

Answer Key for MCQs:

1. (A) [Hint: Use sin²A + cos²A = 1 or draw a triangle with P=3, H=5, find B=4]
2. (B) [Hint: tan 60° = √3, cot 30° = √3]
3. (B) [Hint: Use identity sin²θ + cos²θ = 1]
4. (C) [Hint: Look at the table or divide by cos θ to get tan θ = 1]
5. (B) [Hint: tan 80° = tan(90°-10°) = cot 10°. tan 10° cot 10° = 1]
6. (B) [Hint: Take 9 common, use identity sec²A - tan²A = 1]
7. (C) [Hint: If cos A = 1/2, then sec A = 1/cos A = 2. So, 2 sec A = 2 * 2 = 4]
8. (A) [Hint: 1/2 + 1/2 - 1 = 1 - 1 = 0]
9. (C) [Hint: Check the fundamental identities]
10. (C) [Hint: If tan A = P/B = 4/3, then H = √(P²+B²) = √(4²+3²) = √25 = 5. cosec A = H/P = 5/4]

Study these notes thoroughly and practice the MCQs. Understanding these basics well will make trigonometry much easier going forward. Good luck!