CH – 8 INTRODUCTION TO TRIGONOMETRY
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Master the ratios, conquer the right triangle, and unlock top scores in your board exam!
Chapter Overview
In this chapter, you will learn the fundamental concepts of trigonometry to understand the relationship between angles and sides of right-angled triangles:
- What trigonometry means and how it applies to real-life measurements.
- How to correctly identify sides in right-angled triangles.
- The basic trigonometric ratios: sin, cos, and tan.
- The reciprocal ratios: cosec, sec, and cot.
- Trigonometric ratio values for specific angles: 0°, 30°, 45°, 60°, and 90°.
- Key relationships between trigonometric ratios.
- Standard trigonometric identities and how to prove them.
🎯 Board Exam Focus
Practice the following essential skills carefully to prepare effectively for your board exam:
- Identifying the sides of a right-angled triangle correctly based on the reference angle.
- Clearly understanding the hypotenuse, opposite side, and adjacent side.
- Calculating sin, cos, and tan ratios accurately.
- Finding missing side lengths or trigonometric ratios using the Pythagoras theorem.
- Remembering standard values of trigonometric ratios for 0°, 30°, 45°, 60°, and 90°.
- Applying reciprocal and quotient relationships between ratios.
- Proving trigonometric identities step by step.
Most Important Concepts
What is Trigonometry?
Trigonometry is a branch of mathematics that studies the relationships between the side lengths and angles of a right-angled triangle. The word comes from Greek words: ‘tri’ (three), ‘gon’ (sides), and ‘metron’ (measure). It helps us find unknown heights and distances without measuring them directly.
Understanding a Right-Angled Triangle
A right-angled triangle is a triangle where one angle measures exactly 90°. It contains three main sides:
- Hypotenuse: The longest side of the triangle. It is always the side directly opposite to the 90° angle.
- Opposite Side: The side that lies directly opposite (facing) the acute angle you are looking at.
- Adjacent Side: The side that sits right next to the acute angle you are looking at (forming part of the angle along with the hypotenuse).
C
|\
| \
| \ Hypotenuse
Opposite Side | \ (Longest Side)
(to Angle A) | \
| \
|_____θ\
B A
Adjacent Side
(to Angle A)
• Right angle at B (90°)
• Reference angle is θ at Vertex A
Important Rule: The hypotenuse never changes its position. However, the opposite side and adjacent side change positions depending on which acute angle you pick as your reference angle.
REFERENCE ANGLE: A REFERENCE ANGLE: C
C C
|\ |\
| \ | \
Opp. | \ Hyp. Adj. | \ Hyp.
Side | \ Side | \
(BC) | \ (BC) | \
|_____A |_____A
Adj. Side Opp. Side
(AB) (AB)
Hypotenuse (AC) remains unchanged. Hypotenuse (AC) remains unchanged.
BC is Opposite to Angle A. BC is Adjacent to Angle C.
AB is Adjacent to Angle A. AB is Opposite to Angle C.
Simple Example:
In right-angled triangle ABC, where angle B = 90°:
- The side AC is always the hypotenuse because it faces the 90° angle at B.
- If you look at angle A: Side BC is the opposite side, and side AB is the adjacent side.
- If you switch and look at angle C: Side AB becomes the opposite side, and side BC becomes the adjacent side.
The Six Trigonometric Ratios
Trigonometric ratios are fractions formed by taking the ratio of two sides of a right-angled triangle with respect to an acute angle.
For an acute angle A in a right triangle ABC with right angle at B:
- Sin A (Sine) = Opposite side ÷ Hypotenuse
- Cos A (Cosine) = Adjacent side ÷ Hypotenuse
- Tan A (Tangent) = Opposite side ÷ Adjacent side
- Cosec A (Cosecant) = Hypotenuse ÷ Opposite side
- Sec A (Secant) = Hypotenuse ÷ Adjacent side
- Cot A (Cotangent) = Adjacent side ÷ Opposite side
C
|\
| \
Opposite (Opp) | \ Hypotenuse (Hyp)
| \
|_A__\
Adjacent (Adj)
┌────────────────────────────────────────────────────────┐
│ sin A = Opp / Hyp │ cosec A = Hyp / Opp = 1 / sin A│
│ cos A = Adj / Hyp │ sec A = Hyp / Adj = 1 / cos A│
│ tan A = Opp / Adj │ cot A = Adj / Opp = 1 / tan A│
└────────────────────────────────────────────────────────┘
🧠 Easy Way to Remember Sin, Cos and Tan
Use this simple phrase to memorize the three main ratios:
Some People Have , Curly Brown Hair , Through Proper Brushing
- Some People Have → Sin = Perpendicular (Opposite) ÷ Hypotenuse
- Curly Brown Hair → Cos = Base (Adjacent) ÷ Hypotenuse
- Through Proper Brushing → Tan = Perpendicular (Opposite) ÷ Base
Finding Trigonometric Ratios
Follow these clear steps when solving ratio problems:
- Identify the reference angle mentioned in the question.
- Locate the 90° angle and mark the hypotenuse (side opposite 90°).
- Find the opposite side facing your reference angle.
- Find the adjacent side touching your reference angle.
- Select the required formula and put the side lengths into the fraction.
Example:
In triangle ABC, right-angled at B, AB = 3 cm and BC = 4 cm. Find sin A.
Step 1: Reference angle is angle A.
Step 2: Use Pythagoras theorem to find hypotenuse AC:
AC² = AB² + BC²
AC² = 3² + 4² = 9 + 16 = 25
AC = √25 = 5 cm
Step 3: For angle A, opposite side = BC = 4 cm, hypotenuse = AC = 5 cm.
Step 4: Formula for sin A = Opposite ÷ Hypotenuse = 4 ÷ 5 = 4/5.
⚠️ Common Mistake Warning: Always check your reference angle first! If a question asks for sin C instead of sin A, the opposite side becomes AB instead of BC. Never fix opposite and adjacent sides without checking the angle!
Reciprocal Relationships
Reciprocal means taking a fraction and flipping it upside down. Three trigonometric ratios are direct reciprocals of the other three ratios:
- Cosec A is the reciprocal of Sin A: cosec A = 1 ÷ sin A
- Sec A is the reciprocal of Cos A: sec A = 1 ÷ cos A
- Cot A is the reciprocal of Tan A: cot A = 1 ÷ tan A
Simple Examples:
- If sin A = 3/5, then cosec A = 5/3.
- If cos A = 12/13, then sec A = 13/12.
- If tan A = 4/3, then cot A = 3/4.
Standard Trigonometric Values
Learn the standard values for angles 0°, 30°, 45°, 60°, and 90° clearly:
┌───────────┬─────────┬─────────┬─────────┬─────────┬─────────┐
│ Ratio / θ │ 0° │ 30° │ 45° │ 60° │ 90° │
├───────────┼─────────┼─────────┼─────────┼─────────┼─────────┤
│ sin θ │ 0 │ 1/2 │ 1/√2 │ √3/2 │ 1 │
│ cos θ │ 1 │ √3/2 │ 1/√2 │ 1/2 │ 0 │
│ tan θ │ 0 │ 1/√3 │ 1 │ √3 │ Not Def │
│ cosec θ │ Not Def │ 2 │ √2 │ 2/√3 │ 1 │
│ sec θ │ 1 │ 2/√3 │ √2 │ 2 │ Not Def │
│ cot θ │ Not Def │ √3 │ 1 │ 1/√3 │ 0 │
└───────────┴─────────┴─────────┴─────────┴─────────┴─────────┘
Values for 0°:
- sin 0° = 0
- cos 0° = 1
- tan 0° = 0
- cosec 0° = Not defined
- sec 0° = 1
- cot 0° = Not defined
Values for 30°:
- sin 30° = 1/2
- cos 30° = √3/2
- tan 30° = 1/√3
- cosec 30° = 2
- sec 30° = 2/√3
- cot 30° = √3
Values for 45°:
- sin 45° = 1/√2
- cos 45° = 1/√2
- tan 45° = 1
- cosec 45° = √2
- sec 45° = √2
- cot 45° = 1
Values for 60°:
- sin 60° = √3/2
- cos 60° = 1/2
- tan 60° = √3
- cosec 60° = 2/√3
- sec 60° = 2
- cot 60° = 1/√3
Values for 90°:
- sin 90° = 1
- cos 90° = 0
- tan 90° = Not defined
- cosec 90° = 1
- sec 90° = Not defined
- cot 90° = 0
🧠 Easy Way to Remember Standard Values
Step 1: Finding Sine Values
Write numbers 0, 1, 2, 3, 4.
Divide each by 4: 0/4, 1/4, 2/4, 3/4, 4/4.
Simplify: 0, 1/4, 1/2, 3/4, 1.
Take square roots:
- √0 = 0 (sin 0°)
- √(1/4) = 1/2 (sin 30°)
- √(1/2) = 1/√2 (sin 45°)
- √(3/4) = √3/2 (sin 60°)
- √1 = 1 (sin 90°)
Step 2: Finding Cosine Values
Just reverse the order of Sine values:
- cos 0° = 1
- cos 30° = √3/2
- cos 45° = 1/√2
- cos 60° = 1/2
- cos 90° = 0
Step 3: Finding Tangent Values
Divide sin value by cos value (tan A = sin A ÷ cos A):
- tan 0° = 0 ÷ 1 = 0
- tan 30° = (1/2) ÷ (√3/2) = 1/√3
- tan 45° = (1/√2) ÷ (1/√2) = 1
- tan 60° = (√3/2) ÷ (1/2) = √3
- tan 90° = 1 ÷ 0 = Not defined
Relationships Between Trigonometric Ratios
The following ratios relate directly to each other:
- tan A = sin A ÷ cos A
- cot A = cos A ÷ sin A
- sin A × cosec A = 1
- cos A × sec A = 1
- tan A × cot A = 1
Trigonometric Identities
A trigonometric identity is an equation containing trigonometric ratios that is true for all permissible values of the angle.
The three main NCERT identities are:
- sin² A + cos² A = 1 (valid for 0° ≤ A ≤ 90°)
- 1 + tan² A = sec² A or sec² A − tan² A = 1 (valid for 0° ≤ A < 90°)
- 1 + cot² A = cosec² A or cosec² A − cot² A = 1 (valid for 0° < A ≤ 90°)
PYTHAGOREAN IDENTITIES
Derived from: a² + b² = c² (Pythagoras Theorem)
│
┌────────────────────────────┼────────────────────────────┐
▼ ▼ ▼
sin² A + cos² A = 1 1 + tan² A = sec² A 1 + cot² A = cosec² A
│ │ │
┌────┴────┐ ┌────┴────┐ ┌────┴────┐
│ sin² A │ │ sec² A │ │ cosec² A│
│= 1-cos²A│ │-tan²A= 1│ │-cot²A= 1│
└─────────┘ └─────────┘ └─────────┘
How to use sin² A + cos² A = 1:
- If you need sin² A, write it as: sin² A = 1 − cos² A
- If you need cos² A, write it as: cos² A = 1 − sin² A
🧠 How to Prove a Trigonometric Identity
Use this systematic approach to prove identity questions:
- Choose one side: Start with the Left Hand Side (LHS) or the side that looks more complicated.
- Convert to sin and cos: If terms like tan, cot, sec, or cosec are present, rewrite them using sin and cos.
- Apply known identities: Substitute identities like sin² A + cos² A = 1 or sec² A − tan² A = 1.
- Simplify algebra: Take common terms, simplify fractions using LCM, or expand brackets.
- Reach the target side: Continue simplifying until your result matches the Right Hand Side (RHS).
Board Tip: Do not simplify both LHS and RHS together across an equal sign. Keep LHS on the left, show step-by-step simplification, and end with “= RHS”.
📌 Important Formulas & Results
- sin A = Opposite ÷ Hypotenuse
- cos A = Adjacent ÷ Hypotenuse
- tan A = Opposite ÷ Adjacent
- cosec A = 1 ÷ sin A
- sec A = 1 ÷ cos A
- cot A = 1 ÷ tan A = cos A ÷ sin A
- sin² A + cos² A = 1
- sec² A − tan² A = 1
- cosec² A − cot² A = 1
🧠 How to Know Which Formula or Ratio to Use?
- When to use sin: Use sin when you know or need to find the opposite side and hypotenuse.
- When to use cos: Use cos when you know or need to find the adjacent side and hypotenuse.
- When to use tan: Use tan when you deal only with opposite and adjacent sides (no hypotenuse involved).
- When to use reciprocal ratios: Use cosec, sec, or cot when a given value is already expressed in terms of these ratios.
- When to use identities: Use identities when working with squared terms like sin² A or sec² A, or when proving algebraic trigonometric equations.
🧠 Concepts Students Find Difficult
- Identifying opposite and adjacent sides: Remembering that these sides swap when the reference angle changes.
- Choosing the right ratio: Knowing which ratio connects the given side to the required side.
- Memorizing standard values: Mixing up values such as sin 30° (1/2) and sin 60° (√3/2).
- Understanding reciprocal ratios: Confusing sin with sec instead of cosec.
- Proving identities: Deciding the correct first step in identity proofs.
- Making mistakes with squares: Writing (sin A)² as sin A² instead of sin² A.
⚠️ Common Mistakes Students Make
- Choosing the wrong hypotenuse: Treating the vertical side as the hypotenuse instead of the side facing 90°.
- Fix: Always draw the right angle symbol first and point an arrow to the hypotenuse.
- Mixing opposite and adjacent sides: Assuming the bottom side is always adjacent.
- Fix: Look directly from your reference angle across the triangle to locate the opposite side.
- Forgetting which angle is used: Writing sin A values using side lengths meant for angle C.
- Fix: Mark the given angle clearly before writing any ratio fraction.
- Confusing sin with sec: Thinking sec A is the reciprocal of sin A because both start with different letters.
- Fix: Remember opposites: Sin pairs with Cosec, Cos pairs with Sec.
- Forgetting squares in identities: Writing sin A + cos A = 1 instead of sin² A + cos² A = 1.
- Fix: Identities only apply to squared trigonometric functions.
- Fraction simplification errors: Cancelling terms inside additions incorrectly when simplifying identity fractions.
- Fix: Factorize expressions first before cancelling terms.
- Using incorrect standard values: Swapping tan 30° (1/√3) with tan 60° (√3).
- Fix: Write down the quick sine-cosine memory table in rough work before solving value questions.
🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Proving identities using sin² A + cos² A = 1, sec² A − tan² A = 1, and cosec² A − cot² A = 1.
- Substituting standard values (30°, 45°, 60°) to evaluate expressions.
- Finding all trigonometric ratios given one side ratio and using Pythagoras theorem.
⭐⭐ IMPORTANT PRACTICE
- Finding missing acute angles when ratio values are given.
- Converting tan, cot, sec, and cosec into sin and cos to simplify equations.
- Solving problems involving right triangles with mixed side conditions.
⭐ ADDITIONAL REVISION
- Verifying true/false conceptual statements about ratio limits (e.g., sin A cannot exceed 1).
- Understanding geometric derivations of 30°, 45°, and 60° values.
⚡ Introduction to Trigonometry Quick Revision Sheet
- Right Triangle: Triangle with one 90° angle.
- Sides: Hypotenuse (opposite 90°), Opposite (facing reference angle), Adjacent (next to reference angle).
- Basic Ratios: sin = Opp ÷ Hyp, cos = Adj ÷ Hyp, tan = Opp ÷ Adj.
- Reciprocals: cosec = 1/sin, sec = 1/cos, cot = 1/tan.
- Key Values:
- sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2
- cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2
- tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
- Identities:
- sin² A + cos² A = 1
- sec² A − tan² A = 1
- cosec² A − cot² A = 1
- Quick Reminder: Always check your reference angle before choosing opposite and adjacent sides!
📝 Test Yourself
- In triangle PQR, right-angled at Q, PQ = 5 cm and QR = 12 cm. Find the value of sin P and cos P.
- If cos A = 7/25, calculate the values of tan A and cosec A.
- If 4 cot A = 3, evaluate the value of (sin A + cos A) ÷ (sin A − cos A).
- Evaluate: 2 tan² 45° + cos² 30° − sin² 60°.
- Evaluate: (sin 30° + tan 45°) ÷ (sec 30° + cot 45°).
- If sin (A + B) = 1 and cos (A − B) = √3/2, where 0° < A + B ≤ 90° and A > B, find the values of angle A and angle B.
- Express the ratio sec A in terms of sin A.
- Prove that: (1 − sin A) × (sec A + tan A) = cos A.
- Prove that: tan A + cot A = sec A × cosec A.
- Prove that: (cosec A − sin A) × (sec A − cos A) = 1 ÷ (tan A + cot A).
🎯 Before Your Exam
☐ I can identify a right-angled triangle.
☐ I can identify the hypotenuse correctly.
☐ I can identify opposite and adjacent sides based on the reference angle.
☐ I know the basic meanings and formulas of sin, cos, and tan.
☐ I know all the reciprocal trigonometric ratios.
☐ I remember the standard trigonometric values for 0°, 30°, 45°, 60°, and 90°.
☐ I understand and remember the three main trigonometric identities.
☐ I can prove simple trigonometric identities step by step.
Consistent practice of these core concepts will guarantee complete confidence and clarity in your mathematics board exam!