Basic Concepts of Trigonometry

I. Introduction: Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It has widespread applications in various fields, including physics, engineering, computer science, and more.

Basic Concepts:

I. Angles:

  1. Definition:

    • An angle is formed when two rays share a common endpoint, called the vertex.
  2. Measurement:

    • Angles are measured in degrees () or radians (rad).

II. Right-Angled Triangle:

  1. Definition:

    • A right-angled triangle has one angle that measures exactly 90 degrees.
  2. Sides:

    • The side opposite the right angle is the hypotenuse, and the other two sides are the adjacent side and the opposite side.

III. Trigonometric Ratios:

  1. Sine (sinθ):

    • sinθ=Opposite sideHypotenuse
  2. Cosine (cosθ):

    • cosθ=Adjacent sideHypotenuse
  3. Tangent (tanθ):

    • tanθ=Opposite sideAdjacent side

V. Unit Circle:

  1. Definition:

    • The unit circle is a circle with a radius of 1 centered at the origin in a coordinate plane.
  2. Coordinates:

    • For an angle θ in standard position, the coordinates of the point where the terminal side intersects the unit circle are (cosθ,sinθ).

VI. Trigonometric Identities:

  1. Pythagorean Identity:

    • sin2θ+cos2θ=1
  2. Reciprocal Identities:

    • cscθ=1sinθ
    • secθ=1cosθ
    • cotθ=1tanθ
  3. Quotient and Co-Function Identities:

    • tanθ=sinθcosθ
    • cotθ=cosθsinθ

VII. Trigonometric Functions:

  1. Periodicity:

    • Sine and cosine functions have a period of 2πor 360 degrees.
  2. Graphs:

    • The graphs of sine and cosine functions oscillate between -1 and 1.
  3. Amplitude and Frequency:

    • The amplitude is the maximum value of the function, and the frequency is the number of cycles per unit.