Chapter 8: Introduction to Trigonometry
Understanding Trigonometric Ratios
Trigonometry is the study of relationships between the sides and angles of triangles, especially right-angled triangles. Everyday situations, like estimating the height of a tower or the distance across a river, can be analyzed using right triangles. In a right triangle, specific ratios—sine (sin), cosine (cos), and tangent (tan)—are defined using the lengths of the sides relative to an acute angle. Three more ratios—cosecant (cosec), secant (sec), and cotangent (cot)—are reciprocals of sin, cos, and tan respectively. These ratios are fundamental in solving many geometric problems.
Values and Properties of Trigonometric Ratios
The values of trigonometric ratios depend on the angle, not the size of the triangle. The chapter details the values of these ratios for standard angles: 0°, 30°, 45°, 60°, and 90°, showing how they change as the angle increases. It also explains how to derive trigonometric ratios using geometry and Pythagoras’ theorem, and includes identities such as sin²A + cos²A = 1, 1 + tan²A = sec²A, and 1 + cot²A = cosec²A. These identities are useful for expressing one trigonometric ratio in terms of another.
Applications and Identities
By applying trigonometric identities, one can prove complex expressions and simplify calculations. The chapter includes example problems and formulas that demonstrate how to find unknown angles and sides in triangles using known trigonometric values. It emphasizes that trigonometric equations hold true within specific angle ranges and teaches how to evaluate, manipulate, and transform expressions using identities. These skills are foundational for further studies in geometry, physics, and engineering.