Notes on Trigonometric Functions

3.1 Introduction to Trigonometric Functions

• Trigonometry is derived from Greek, meaning "measuring the sides of a triangle." It originally arose to solve problems involving triangles. Key historical applications include navigation by sea captains, land surveying, and engineering.
• Modern Applications: Today, trigonometry plays a significant role in diverse areas such as seismology, electrical circuit design, atomic physics, and wave analysis (e.g., musical tones).

3.2 Understanding Angles

• Angle Definition: An angle measures the rotation between an initial and terminal side around a vertex. Positive angles result from anticlockwise rotation, while negative angles come from clockwise rotation.
• **Units of Measurement:
  1. Degree Measure: When an angle results from a rotation of 1/360 of a complete revolution, it is 1 degree (1°). There are 60 minutes in a degree (1′) and 60 seconds in a minute (1″).
  2. Radian Measure: One radian is the angle subtended at the center of a unit circle by an arc length equal to the radius. There are 2π radians in one full revolution (360°) and π radians in 180°.
• Conversion between degrees and radians:
  • 1 radian = 180/π degrees
  • 1 degree = π/180 radians
  • Notable conversions:
    • 0° = 0 rad
    • 30° = π/6 rad
    • 45° = π/4 rad
    • 60° = π/3 rad
    • 90° = π/2 rad
    • 180° = π rad
    • 360° = 2π rad
• Real-Number Association: Each angle corresponds to a real number in a complete rotation around the unit circle, linking radians directly to real numbers.

3.3 Trigonometric Functions Defined

• Basic Definitions Using the Unit Circle: In a unit circle:
  • cos(x) is the x-coordinate of a point on the circle,
  • sin(x) is the y-coordinate.
• Identity: The relationship between sine and cosine follows the identity:
  [ cos^2(x) + sin^2(x) = 1 ]
• Trigonometric Functions for Any Angle: Trigonometric functions can be extended beyond acute angles (0° to 90°) to include all angles in the unit circle framework.
• Reciprocal Functions:
  • cosec(x) = 1/sin(x)
  • sec(x) = 1/cos(x)
  • tan(x) = sin(x)/cos(x)
  • cot(x) = cos(x)/sin(x)

3.3.1 Sign of Trigonometric Functions

• Quadrants:
  • 1st Quadrant (0° to 90°): sin and cos are positive.
  • 2nd Quadrant (90° to 180°): sin is positive, cos is negative.
  • 3rd Quadrant (180° to 270°): sin and cos are negative.
  • 4th Quadrant (270° to 360°): sin is negative, cos is positive.
• Understanding the sign of the trigonometric functions in different quadrants allows for easier calculations and expectations of their values based on the angle's location.

3.3.2 Domain and Range

• The functions sin(x) and cos(x) are defined for all real numbers, both ranging from -1 to 1. The functions tan(x), cot(x), sec(x), and cosec(x) are restricted in their domains where their respective denominators are not zero.

3.4 Trigonometric Functions of Sum and Difference of Angles

• The identities for the sine and cosine of angle sums and differences enable simplification of calculations:
  • cos(x ± y) = cos x cos y ∓ sin x sin y
  • sin(x ± y) = sin x cos y ± cos x sin y

Example Calculations

• Given:
  If tan(x) = a, find sin(x), cos(x):
  • sin = a/√(1+a²)
  • cos = 1/√(1+a²)

Summary of Key Identities

1. Angle Measurements: Relationship between degrees and radians.
2. Pythagorean Identity: [ cos^2(x) + sin^2(x) = 1 ]
3. Even-Odd Properties:
   • sin(-x) = -sin x; cos(-x) = cos x
4. Additive Formulas:
   • sin(x ± y) = sin x cos y ± cos x sin y
   • cos(x ± y) = cos x cos y ∓ sin x sin y
5. Tangent Relationships:
   • tan(x) = sin(x)/cos(x)
   • tan(x ± y) = (tan x ± tan y)/(1 ∓ tan x tan y)

Historical Context

• Trigonometry began in ancient India with figures like Aryabhata and Brahmagupta contributing essential concepts and formulas. The Indian methods later influenced the mathematical practices in the Middle East and ultimately Europe.

Learning about trigonometric functions equips students with essential mathematical tools applicable in numerous real-world scenarios, thus laying the groundwork for advanced studies in mathematics and its applications in sciences.

1. Trigonometry involves measuring the sides of triangles.
2. Angles can be measured in degrees or radians.
3. The relationship of sine and cosine is expressed as: [ cos^2(x) + sin^2(x) = 1 ]
4. Trigonometric functions can apply to any angles, not just acute.
5. The unit circle is fundamental in defining sine and cosine values.
6. In different quadrants, sine and cosine functions have varying signs.
7. The domain of sine and cosine functions is all real numbers, while tangent has specific limits.
8. Key sum and difference identities for trigonometric functions aid in calculations.
9. Historical development of trigonometry traces back to ancient Indian mathematicians significantly influencing modern mathematics.