Definition and Meaning of Sine and Cosine - Part One

Overview

• Most important lesson for trigonometry across algebra, pre-calculus, and calculus.
• Focus on understanding what sine and cosine really mean rather than just memorizing definitions.
• Aim to internalize the concepts, similar to learning the alphabet before writing.

The Concept of Force and Angles

• Example scenario: pushing a box on the ground.
• Pushing with a force (in Newtons) at an angle creates a need to break the force into components.
  • Force: Example of pushing with 5 Newtons.
  • Force can be broken into horizontal and vertical components.
  • Draw a triangle to visualize the components.

Right Triangles and Special Triangles

• Focus on right triangles (90-degree angle).
• Special triangle example: 3-4-5 triangle.
  • If hypotenuse = 5, then the other sides are:
    • Vertical component: 3 (opposite)
    • Horizontal component: 4 (adjacent)
• Pythagorean theorem confirms that these sides form a right triangle.

Breaking Forces into Components

• Purpose of sine and cosine: to break the angled force into horizontal and vertical components.
• Key takeaway:
  • Sine function gives the vertical component.
  • Cosine function gives the horizontal component.

Definition of Sine and Cosine

• Sine Definition:

[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6 ]

• Indicates how much force is in the vertical direction compared to total force.

• Cosine Definition:

[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8 ]

• Indicates how much force is in the horizontal direction compared to total force.

Understanding Projections

• Sine corresponds to the vertical projection (y-direction).
• Cosine corresponds to the horizontal projection (x-direction).
• Remember: Sine = y, Cosine = x.

Calculating Force Components

• Sine and cosine provide ratios less than 1:
  • 0.6 indicates 60% of the total force is in the y-direction.
  • 0.8 indicates 80% of the total force is in the x-direction.
• Sine and cosine can be viewed as chopping functions:
  • Sine chops the hypotenuse down for the y-direction.
  • Cosine chops the hypotenuse down for the x-direction.

Application to Different Forces

• Example of adjusting force:
  • Pushing with 12 Newtons at the same angle (36.87 degrees).
  • Calculate vertical and horizontal components:
    • Vertical (y): [ 12 \times 0.6 = 7.2 \text{ Newtons} ]
    • Horizontal (x): [ 12 \times 0.8 = 9.6 \text{ Newtons} ]

Verifying the Calculation

• Use Pythagorean theorem to verify calculated sides form a right triangle:
  • Example: 12² = 7.2² + 9.6².

Importance of Sine and Cosine

• Sine and cosine are critical for breaking down vectors (forces, velocities, etc.) into components for easier calculations.
• They serve as foundational concepts for understanding more complex mathematical and physical concepts later in studies.

Closing Thoughts

• Revisit key concepts to internalize for future application in trigonometry and calculus.
• Encourage multiple reviews to solidify understanding.