Introduction to Trigonometry (Class 10): Notes और MCQ Practice

आसान भाषा में:

Trigonometry का मतलब है Right-angled triangle की sides और angles का रिश्ता समझना। इसमें एक acute angle (θ) के reference से sides के ratios निकाले जाते हैं, जिन्हें Trigonometric Ratios कहते हैं। यह chapter board exam के लिए super scoring है!

Important Formulas & Points:

• T-Ratios (Right Triangle में):
• sin θ = Perpendicular ÷ Hypotenuse (P ÷ H)
• cos θ = Base ÷ Hypotenuse (B ÷ H)
• tan θ = Perpendicular ÷ Base (P ÷ B) = sin θ ÷ cos θ
• cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ

• Standard Values (0°, 30°, 45°, 60°, 90°):
• sin: 0, 1/2, 1/√2, √3/2, 1
• cos: 1, √3/2, 1/√2, 1/2, 0 (sin का उल्टा order)
• tan: 0, 1/√3, 1, √3, Not Defined

• 3 Main Identities:

1. sin² θ + cos² θ = 1
2. 1 + tan² θ = sec² θ
3. 1 + cot² θ = cosec² θ

2 Super Memory Tricks:

1. P/H, B/H, P/B याद रखने का देसी मंत्र:

"Pandit Badri Prasad, Har Har Bole"

(P/H = sin, B/H = cos, P/B = tan).

1. Table बनाने की Hand Trick:

0°, 30°, 45°, 60°, 90° के लिए 0, 1, 2, 3, 4 लिखें, सबको 4 से divide करें और root ले लें: √(0/4), √(1/4), √(2/4), √(3/4), √(4/4)। इससे सीधे sin की values मिल जाएंगी!

2 Common Board Exam Mistakes:

1. Base और Perpendicular में Confusion: हमेशा याद रखें, जिस angle (θ) की बात हो रही है, उसके सामने वाली side ही Perpendicular होती है और साथ वाली Base। Angle बदलते ही P और B swap हो जाते हैं!
2. Algebraic Multiplication का भ्रम: sin A का मतलब sin × A नहीं होता। इसलिए sin(A + B) कभी भी sin A + sin B के बराबर नहीं होता!

Class 10 Introduction to Trigonometry - Study Notes

1. Chapter in Short

Trigonometry comes from three Greek words: tri (meaning three), gon (meaning sides), and metron (meaning measure).

This chapter teaches how the angles and sides of a right-angled triangle are related to each other. Instead of directly climbing a tall tower or crossing a wide river to measure distances, trigonometry allows us to calculate unknown heights and distances using angles and known side lengths.

2. Key Concepts Explained Simply

The Sides of a Right-Angled Triangle

In any right-angled triangle, the three sides are named with respect to a chosen acute angle (let us call it angle A or θ):

• Hypotenuse: The longest side, always directly opposite to the 90° right angle.
• Opposite side (Perpendicular): The side directly facing the angle you are considering.
• Adjacent side (Base): The side sitting next to the angle you are considering (touching both the angle and the right angle).

Everyday Example:

Imagine you are standing on flat ground looking up at the top of the Qutub Minar.

• The ground between your feet and the base of the tower is the Adjacent side.
• The height of the tower is the Opposite side.
• Your direct line of sight from your eyes to the top of the tower forms the Hypotenuse.

The Six Trigonometric Ratios

Trigonometric ratios are simply fractions comparing two sides of a right triangle:

1. Sine (sin): Compares the opposite side to the hypotenuse.
2. Cosine (cos): Compares the adjacent side to the hypotenuse.
3. Tangent (tan): Compares the opposite side to the adjacent side.
4. Cosecant (cosec): The reciprocal (flip) of sine.
5. Secant (sec): The reciprocal (flip) of cosine.
6. Cotangent (cot): The reciprocal (flip) of tangent.

Important Note: The symbol sin A is an abbreviation for "sine of angle A". It does not mean "sin multiplied by A". Without an angle, "sin" has no mathematical meaning.

3. Important Definitions, Formulas, and Values

Basic Ratios (for acute angle A)

• sin A = Opposite side ÷ Hypotenuse
• cos A = Adjacent side ÷ Hypotenuse
• tan A = Opposite side ÷ Adjacent side
• cosec A = Hypotenuse ÷ Opposite side = 1 ÷ sin A
• sec A = Hypotenuse ÷ Adjacent side = 1 ÷ cos A
• cot A = Adjacent side ÷ Opposite side = 1 ÷ tan A

Quotient Relations

• tan A = sin A ÷ cos A
• cot A = cos A ÷ sin A

Trigonometric Values Table

Ratio — 0° — 30° — 45° — 60° — 90°

sin — 0 — 1/2 — 1/√2 — √3/2 — 1

cos — 1 — √3/2 — 1/√2 — 1/2 — 0

tan — 0 — 1/√3 — 1 — √3 — Not Defined

cosec — Not Defined — 2 — √2 — 2/√3 — 1

sec — 1 — 2/√3 — √2 — 2 — Not Defined

cot — Not Defined — √3 — 1 — 1/√3 — 0

Fundamental Trigonometric Identities

These equations are true for all allowable values of angle A:

1. sin² A + cos² A = 1

(can be rewritten as: sin² A = 1 − cos² A or cos² A = 1 − sin² A)

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

(can be rewritten as: sec² A − tan² A = 1)

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

(can be rewritten as: cosec² A − cot² A = 1)

(Note: sin² A means (sin A)²)

4. Solved Examples (Board-Exam Style)

Example 1: Finding Ratios Using a Triangle

Question: In a right triangle ABC, right-angled at B, if tan A = 4/3, find sin A and cos A.

Solution:

1. By definition, tan A = Opposite side ÷ Adjacent side = BC ÷ AB = 4/3.
2. Let BC = 4k and AB = 3k, where k is a positive number.
3. Using Pythagoras theorem in triangle ABC:

AC² = AB² + BC²

AC² = (3k)² + (4k)²

AC² = 9k² + 16k² = 25k²

AC = √(25k²) = 5k (Hypotenuse)

1. Now calculate the required ratios:

• sin A = Opposite ÷ Hypotenuse = BC ÷ AC = 4k ÷ 5k = 4/5
• cos A = Adjacent ÷ Hypotenuse = AB ÷ AC = 3k ÷ 5k = 3/5

Example 2: Evaluating Numerical Expressions

Question: Evaluate the value of: 2 tan² 45° + cos² 30° − sin² 60°.

Solution:

1. Substitute known standard values from the table:

• tan 45° = 1
• cos 30° = √3/2
• sin 60° = √3/2

1. Plug these values into the expression:

= 2 × (1)² + (√3/2)² − (√3/2)²

= 2 × 1 + 3/4 − 3/4

= 2 + 0

= 2

Example 3: Finding Unknown Angles

Question: If sin (A − B) = 1/2 and cos (A + B) = 1/2, where 0° < (A + B) ≤ 90° and A > B, find the values of A and B.

Solution:

1. From the values table, we know sin 30° = 1/2.

Therefore, A − B = 30° --- (Equation 1)

1. We also know cos 60° = 1/2.

Therefore, A + B = 60° --- (Equation 2)

1. Add Equation 1 and Equation 2:

(A − B) + (A + B) = 30° + 60°

2A = 90°

A = 45°

1. Substitute A = 45° into Equation 2:

45° + B = 60°

B = 60° − 45° = 15°

Answer: A = 45° and B = 15°

Example 4: Proving an Identity

Question: Prove that: (sin θ − 2 sin³ θ) ÷ (2 cos³ θ − cos θ) = tan θ.

Solution:

1. Take the Left Hand Side (LHS):

LHS = (sin θ − 2 sin³ θ) ÷ (2 cos³ θ − cos θ)

1. Take common terms out from numerator and denominator:

Numerator: sin θ × (1 − 2 sin² θ)

Denominator: cos θ × (2 cos² θ − 1)

LHS = [sin θ × (1 − 2 sin² θ)] ÷ [cos θ × (2 cos² θ − 1)]

1. Use the identity 1 = sin² θ + cos² θ in the numerator bracket:

1 − 2 sin² θ = (sin² θ + cos² θ) − 2 sin² θ = cos² θ − sin² θ

1. Use the identity 1 = sin² θ + cos² θ in the denominator bracket:

2 cos² θ − 1 = 2 cos² θ − (sin² θ + cos² θ) = cos² θ − sin² θ

1. Substitute these back into LHS:

LHS = [sin θ × (cos² θ − sin² θ)] ÷ [cos θ × (cos² θ − sin² θ)]

1. The term (cos² θ − sin² θ) cancels out:

LHS = sin θ ÷ cos θ = tan θ = RHS

Hence proved.

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid

1. Swapping Opposite and Adjacent sides:

• Always identify the angle first. If you are working with angle A, the opposite side is the side in front of A. If you switch to angle C, the opposite side changes to AB.

1. Confusing (sin A)² with sin A²:

• (sin A)² is written as sin² A. It means the ratio is squared.
• It is not equal to sin(A²).

1. Assuming values can be anything:

• The value of sin A and cos A can never be greater than 1 because the hypotenuse is always the longest side, so Opposite ÷ Hypotenuse is always ≤ 1.

1. Division by zero:

• tan 90° and sec 90° are "Not Defined" because their denominator (cos 90°) is 0.

Quick Memory Tips

• Easy trick to remember the 30°, 45°, 60° values:

Write numbers 0, 1, 2, 3, 4 under 0°, 30°, 45°, 60°, 90°.

Divide all by 4 and take the square root:

• √(0/4) = 0 (sin 0°)
• √(1/4) = 1/2 (sin 30°)
• √(2/4) = 1/√2 (sin 45°)
• √(3/4) = √3/2 (sin 60°)
• √(4/4) = 1 (sin 90°)

To get cos, simply reverse the sequence of numbers!