HSSLIVE Plus One Maths Chapter 3: Trigonometric Functions Notes

Trigonometric functions bridge geometry and analysis, providing essential tools for describing periodic phenomena. This chapter covers the measurement of angles, the six trigonometric functions and their relationships, trigonometric identities, and equations. Students will develop proficiency in manipulating these functions and applying them to solve problems involving triangles, circular motion, and wave patterns. These skills provide a foundation for advanced calculus, physics, engineering, and many natural sciences.

Chapter 3: Trigonometric Functions

Angles and Their Measurement:

• Degree Measure: 360° in a complete rotation
• Radian Measure: 2π radians in a complete rotation
• Conversion: 180° = π radians, or θ(in radians) = π/180 × θ(in degrees)

Trigonometric Ratios in Right-Angled Triangles:

• sin θ = Opposite/Hypotenuse
• cos θ = Adjacent/Hypotenuse
• tan θ = Opposite/Adjacent = sin θ/cos θ
• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ = cos θ/sin θ

Trigonometric Functions of Any Angle:

Using the unit circle (x = cos θ, y = sin θ), we define:

• sin θ = y-coordinate
• cos θ = x-coordinate
• tan θ = y/x (x ≠ 0)

Signs of Trigonometric Functions in Different Quadrants:

• First Quadrant (0° to 90°): All positive
• Second Quadrant (90° to 180°): Only sin, cosec positive
• Third Quadrant (180° to 270°): Only tan, cot positive
• Fourth Quadrant (270° to 360°): Only cos, sec positive

Important Values:

Fundamental Trigonometric Identities:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ
4. sin(A + B) = sinA·cosB + cosA·sinB
5. cos(A + B) = cosA·cosB – sinA·sinB
6. tan(A + B) = (tanA + tanB)/(1 – tanA·tanB)

Addition and Subtraction Formulas:

• sin(A – B) = sinA·cosB – cosA·sinB
• cos(A – B) = cosA·cosB + sinA·sinB
• tan(A – B) = (tanA – tanB)/(1 + tanA·tanB)

Double Angle Formulas:

• sin(2A) = 2sinA·cosA
• cos(2A) = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A
• tan(2A) = 2tanA/(1 – tan²A)