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Chapter Analysis
Intermediate16 pages • EnglishQuick Summary
Chapter 8 introduces the concept of trigonometry, focusing on trigonometric ratios in right-angled triangles, including sine, cosine, and tangent. It defines these ratios for specific angles such as 30°, 45°, 60° and 90°. The chapter also covers trigonometric identities and relationships, demonstrating how to derive various trigonometric values from basic ones.
Key Topics
- •Trigonometric Ratios
- •Trigonometric Functions
- •Applications of Trigonometry
- •Trigonometric Identities
- •Specific Angle Trigonometry
- •Defining Angles Using Trigonometry
- •Relationship of Ratios
Learning Objectives
- ✓Understand the definitions of trigonometric ratios.
- ✓Calculate trigonometric function values for specific angles.
- ✓Prove basic trigonometric identities.
- ✓Apply identities to determine unknown sides or angles in right triangles.
- ✓Comprehend the relationships between trigonometric functions.
Questions in Chapter
In ∆ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: (i) sin A, cos A (ii) sin C, cos C
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If sin A = 3/4, calculate cos A and tan A.
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Given 15 cot A = 8, find sin A and sec A.
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If tan (A + B) = 3 and tan (A – B) = 1/3; 0° < A + B ≤ 90°; A > B, find A and B.
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State whether the following are true or false. Justify your answer. (i) sin(A + B) = sin A + sin B.
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Additional Practice Questions
Prove that if tan A = 1, then sin^2 A + cos^2 A = 1.
mediumAnswer: In a right triangle where tan A = 1, both adjacent and opposite sides must be equal. Hence, sin A = cos A = 1/√2. Therefore, sin^2 A + cos^2 A = 1/2 + 1/2 = 1.
Evaluate the expression: sin^2 45° + cos^2 45°.
mediumAnswer: sin 45° = cos 45° = 1/√2. Thus, sin^2 45° + cos^2 45° = (1/√2)^2 + (1/√2)^2 = 1/2 + 1/2 = 1.
Determine the value of sin 0° and cos 0° using trigonometric definitions.
easyAnswer: As angle approaches 0, the opposite side becomes negligible compared to the hypotenuse, hence sin 0° = 0/1 = 0. The adjacent side becomes equal to the hypotenuse making cos 0° = 1/1 = 1.
If sec A = 13/12, evaluate tan A.
hardAnswer: Using identity sec^2 A = 1 + tan^2 A, we get (13/12)^2 = 1 + tan^2 A. Hence, tan^2 A = (169/144) - 1 = 25/144, thus tan A = 5/12.
Express the trigonometric ratios: sin A, sec A, and tan A in terms of cot A.
mediumAnswer: sin A = 1/√(1 + cot^2 A), sec A = √(1 + cot^2 A), tan A = 1/cot A.