Introduction to Trigonometry — الرياضيات – الصف 10 (CBSE) بشرح مبسّط

ما ستتعلّمه

1. 1

   الدرس 1: Right Triangles and Well-Defined Ratios

   6 دقائق
   • Right Triangles Connect Heights, Distances and Angles
   • Opposite and Adjacent Depend on the Reference Angle
   • Define the Three Basic Trigonometric Ratios
   • Connect All Six Ratios
   • Enrichment: How Sine and Cosine Get Their Namesغير مُقيَّم
   • Similarity Makes the Ratios Well Defined
   • Read Ratio Names and Squares Correctly
   • Enrichment: A Reciprocal Is Not Sine Inverse
   • A Known Ratio Determines a Missing Side Ratio
2. 2

   الدرس 2: Using Ratios and Building Standard-Angle Triangles

   6 دقائق
   • Example 1: Find the Other Five Ratios
   • Example 2: Equal Sines Give Equal Acute Angles
   • Example 3: Evaluate a Sum and Difference of Squares
   • Example 4: Verify the Given Product
   • Example 5: Use a Difference of Side Lengths
   • Exercise 8.1: Interpret Before Calculating
   • Derive the Six Ratios for 45°
   • An Equilateral Triangle Produces 30° and 60°
3. 3

   الدرس 3: Standard Angles and Their Applications

   6 دقائق
   • Read the Ratios for 30° and 60°
   • Motivate the Ratios at 0°
   • Motivate the Ratios at 90°
   • Interpret Table 8.1
   • Example 6: Select Ratios for Missing Sides
   • Example 7: Determine the Acute Angles
   • Example 8: Solve for A and B
   • Exercise 8.2: Substitute and Justify
4. 4

   الدرس 4: Trigonometric Identities and Practice

   8 دقائق
   • Prove the Sine–Cosine Identity
   • Derive the Tangent–Secant Identity
   • Derive the Cotangent–Cosecant Identity
   • Find Ratios Using Identities
   • Example 9: Express Ratios in Terms of Sine
   • Example 10: Convert to Sine and Cosine
   • Example 11: Factor Before Cancelling
   • Enrichment: Follow the Longer Proof in Example 12
   • Exercise 8.3: Choose a Short Identity Route
   • Recap: Definitions, Values, Identities and Practice

المفاهيم الأساسية

Right triangles in surroundings

Trigonometry studies relationships between the sides and angles of a triangle. Its name comes from Greek words meaning three, sides and measure.

Optional inverse-notation distinction

The expression called sine inverse A is different. It is not another way to write cosecant A.

Recovering a ratio geometrically

A side length is positive, so choose the positive value.

Forty-five-degree values

Cosine forty five degrees has the same value.

Zero-degree ratios

Cotangent and cosecant zero degrees are not defined because their reciprocal expressions divide by zero. The endpoint values are definitions motivated by the changing triangle.

Ninety-degree ratios

At ninety degrees, both require division by zero, so neither is defined.

Fundamental trigonometric identity

An identity is an equation true for all values under consideration.

Cotangent–cosecant identity

A valid identity must use expressions that are defined.

تركيز الامتحان

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

INTRODUCTION TO TRIGONOMETRY: 1. Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined). 2. Motivate the ratios whichever are defined at 0° and 90°. Values of the trigonometric ratios of 30°, 45° and 60°. 3. Relationships between the ratios. TRIGONOMETRIC IDENTITIES: 1. Proof and applications of the identity sin^2 A + cos^2 A = 1. 2. Only simple identities to be given. Competencies: understands the definitions of the basic trigonometric functions (including the introduction of the sine and cosine functions); uses trigonometric identities to solve problems. Learning outcomes: evaluates trigonometric ratios; describes trigonometric ratios of standard angles and solving related expressions; proves trigonometric identities using sin^2 A + cos^2 A = 1 and other identities.

Not assessed

These parts go beyond the 2026-27 syllabus and are taught for understanding only: Enrichment: How Sine and Cosine Get Their Names.

Exam note

This one textbook chapter covers two syllabus topics: Introduction to Trigonometry and Trigonometric Identities (8.4).

Exam note

Identities: "Only simple identities to be given" - keep identity proofs short and simple.

Exam note

Trigonometric ratios of complementary angles are not in the syllabus (the textbook mentions complementary angles only in its historical note).

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

118 board-pattern questions with marking schemes: 33 multiple choice (1 mark), 15 assertion–reason (1 mark), 26 very short answer (2 marks), 25 short answer (3 marks), 6 long answer (5 marks) and 13 case-based (4 marks).

أسئلة نموذجية مع نقاط التصحيح

أسئلة بنمط امتحان المجلس من مجموعة تدريبات AI Lessons لهذا الفصل. توضّح نقاط التصحيح ما يبحث عنه المصحّح؛ والإجابات النموذجية الكاملة في التطبيق.

1. Very short answerدرجتان

   Given sin A = 1/3, a student calculates sin² A as 2/3. Identify the error and find the correct value.

   نقاط التصحيح

   • Identifies that sin² A means (sin A)², not 2 sin A.1
   • Correctly squares 1/3 to obtain 1/9.1

2. Short answer3 درجات

   A student writes: 'Since tan A = 4/3, sin A = 4/3 and cos A = 3/4.' Explain the errors and determine the correct values of sin A, cos A and cot A.

   نقاط التصحيح

   • Explains that sine and cosine require the hypotenuse, while 3/4 is the reciprocal of tangent.0.5
   • Uses proportional legs 4k and 3k and Pythagoras to obtain hypotenuse 5k.1
   • Obtains sin A = 4/5.0.5
   • Obtains cos A = 3/5.0.5
   • Obtains cot A = 3/4.0.5

3. Long answer5 درجات

   For an acute angle A, sec A = 13/12. A student claims that (1 + tan² A)/(1 + cot² A) = 144/25 and (1 - tan A)/(1 - cot A) = 5/12. Check both claims using identities and reciprocal relationships. Give the correct values and explain why the second quotient is defined for this angle.

   نقاط التصحيح

   • Uses tan² A = sec² A - 1 to obtain 25/144.1
   • Chooses the positive value tan A = 5/12.0.5
   • Uses the reciprocal relationship to obtain cot A = 12/5.0.5
   • Simplifies the first quotient to tan² A = 25/144 using identities, or obtains the same value by correct substitution.1
   • Shows that the second denominator is -7/5 ≠ 0.0.5
   • Substitutes correctly into the second quotient to obtain (7/12)/(-7/5).0.5
   • Obtains the second quotient as -5/12.0.5
   • Explicitly rejects both proposed values.0.5

الأسئلة الشائعة

فصول أخرى في الرياضيات – الصف 10

• الفصل 1Real Numbersالدروس: 4 · الدقائق: 21→
• الفصل 2Polynomialsالدروس: 4 · الدقائق: 21→
• الفصل 3Pair of Linear Equations in Two Variablesالدروس: 4 · الدقائق: 24→
• الفصل 4Quadratic Equationsالدروس: 4 · الدقائق: 22→

كل أدلة الرياضيات – الصف 10 (CBSE) →