Introduction to Trigonometry

Context and Introduction

• The speaker opens with an anecdotal comment about inappropriate behavior in an educational context.

Transition to Geometry and Trigonometry

• The discussion shifts to triangles, specifically in the context of trigonometry.

• Key Questions in Trigonometry:

  • Can we make a triangle?

  • Can we make a right triangle?

  • Can we represent it using a circle?

Right Triangles in Trigonometry

• Definition of a Right Triangle:

  • A triangle with one angle measuring 90 degrees.

• Number of Right Angles in a Triangle:

  • A triangle can have only one right angle.

• Example of Practical Visualization:

  • Cutting an orange into eight triangular pieces, each with three right angles.

• Terminology:

  • Side opposite the right angle: Hypotenuse (h)

    • Definition: The longest side in the right triangle.

  • The two other sides that form the right angle are called Legs (l₁ and l₂).

Naming Angles in a Triangle

• Selecting one acute angle, denoted as Theta (θ):

  • The side opposite the angle is termed the Opposite Side (o).

  • The side adjacent to the angle is termed the Adjacent Side (a).

• Established Naming Convention:

  • The identification of opposite and adjacent sides depends on which acute angle (θ) is selected.

Trigonometric Ratios

• Importance of Trigonometric Ratios:

  • The relationship between angles and sides in right triangles.

• Key Ratios Derived from θ:

  • Sine (sin θ):

    • ext{sin θ} = rac{ ext{o}}{ ext{h}}

  • Cosine (cos θ):

    • ext{cos θ} = rac{ ext{a}}{ ext{h}}

  • Tangent (tan θ):