Introduction to Trigonometric Functions

Introduction to Trigonometric Functions

• Trigonometric functions relate angles in a right-angled triangle to ratios of side lengths.
• These functions are fundamental in mathematics, physics, and engineering.
• Common trigonometric functions include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Definitions in a Right-Angled Triangle

• Consider a right-angled triangle with:

  • Angle θ
  • Opposite side (opposite to θ)
  • Adjacent side (adjacent to θ and excluding the hypotenuse)
  • Hypotenuse (the longest side, opposite the right angle)

• Sine (sin θ) = Opposite / Hypotenuse

• Cosine (cos θ) = Adjacent / Hypotenuse

• Tangent (tan θ) = Opposite / Adjacent

Definitions Using the Unit Circle

• Define the trigonometric functions using the unit circle approach.
• The unit circle is a circle with a radius of 1 centered at the origin (0,0) of a Cartesian plane.
• An angle θ in standard position has its initial side on the positive x-axis.
• The point where the terminal side intersects the unit circle has coordinates (cos θ, sin θ).

Relationships Between Trigonometric Functions

• Reciprocal Identities:

  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ = cos θ / sin θ

• Pythagorean Identities:

  • sin² θ + cos² θ = 1

• Tangent Identity:

  • tan θ = sin θ / cos θ

Trigonometric Functions of Special Angles

• Knowledge of trigonometric functions for special angles (0°, 30°, 45°, 60°, 90°) is crucial.
• These values are often memorized or readily derived from geometric relationships in special triangles.

Graphs of Trigonometric Functions

• Sine function (sin x): periodic function that oscillates between -1 and 1, with a period of 2π.
• Cosine function (cos x): similar to the sine function, but shifted horizontally by π/2.
• Tangent function (tan x): periodic function with vertical asymptotes at odd multiples of π/2.

Domain and Range of Trigonometric Functions

• Domain: the set of all possible input values (angles) for a function.
• Range: the set of all possible output values for a function.
• The domains and ranges are defined differently for each trigonometric function and can include restrictions.

Applications of Trigonometric Functions

• Trigonometry is used in navigation, surveying, engineering, and many other fields.
• Essential for modeling periodic phenomena, particularly in areas like sound and light waves, and harmonic motion.

Unit Circle and Angles

• The unit circle is a valuable tool for visualizing trigonometric values for any angle.
• Angles can be measured in degrees or radians.
• Conversion between degree and radian measure is important.