Graphing the Sine Function using the Unit Circle

Representation of Circles and Trigonometric Functions as Infinite Sets of Ordered Pairs

  • The Unit Circle as a Collection of Continuous Points:
    • A circle is defined mathematically as an infinite collection of individual points or ordered pairs.
    • Although a circle appears to the naked eye as a smooth, continuous line, it consists of an infinite number of points laid out completely around its perimeter.
    • Placing a marker at any position along the circle identifies a distinct coordinate pair.
    • Specific coordinate examples on the unit circle include:
    • (−0.6,0.8)(-0.6, 0.8)
    • (−1,0)(-1, 0)
  • Graphing Trigonometric Functions:
    • Graphing functions such as sine (sin⁡(x)\sin(x)) or cosine (cos⁡(x)\cos(x)) relies on finding an infinite number of ordered pairs.
    • Just like a circle, the graph of a sine or cosine function consists of ordered pairs (x,y)(x, y).
    • The primary objective when graphing these functions is to determine and plot all ordered pairs representing the function across the coordinate space.

Measuring Angles and Constructing the Sine Function

  • Measurement of Angles in Radians:
    • Angles along the horizontal axis are measured in radians.
    • Radians are defined as unitless numbers.
  • Conceptual Setup for Graphing y=sin⁡(x)y = \sin(x):
    • When evaluating y=sin⁡(x)y = \sin(x), the equation can be expressed as y=sin⁡(θ)y = \sin(\theta), where:
    • The horizontal axis (xx-axis) represents the angle θ\theta measured in unitless radians.
    • The vertical axis (yy-axis) represents the height of a reference triangle within the unit circle, corresponding to sin⁡(θ)\sin(\theta).
  • Initial State at Zero Radians:
    • When a reference triangle is constructed extremely low near the horizontal axis, the angle θ\theta is exceptionally small.
    • If the angle θ=0\theta = 0, the vertical height of the triangle is equal to 00
    • This establishes the first exact ordered pair on the graph of the sine function:
    • (0,0)(0, 0)

Behavior of the Sine Graph in Quadrant I

  • Continuous Progression of Angle and Height:
    • As the angle θ\theta increases to a small positive value, a continuous geometric process occurs.
    • Moving to the right along the horizontal axis (increasing θ\theta) causes the height of the reference triangle to increase (increasing yy).
    • As the triangle expands, higher values for both the angle θ\theta and vertical height yy produce additional ordered pairs progressing upward and to the right on the graph.
  • Transition Toward the Top of the Circle:
    • As the process continues, θ\theta becomes a large angle within the first quadrant, and the vertical height yy increases toward the maximum height at the top of the circle.

Questions & Discussion

  • Identifying the Angle at the Top of the Unit Circle:
    • Question: What angle θ\theta is being approached as the reference triangle moves from 00 to the top of the circle?
    • Incorrect Proposals:
    • 490490
    • 11
    • Correct Answer in Degrees:
    • 9090 degrees (90o90^\text{o}
    • Correct Answer in Radians:
    • π2\frac{\pi}{2} radians
    • Graphical Significance of π2\frac{\pi}{2}:
    • The value π2\frac{\pi}{2} corresponds to the peak of the unit circle.
    • The horizontal coordinate on the sine graph approaches approximately π2\frac{\pi}{2} as the reference triangle reaches the top of the circle.
  • Maximum Height Evaluation:
    • Question: What is the tallest possible height (yy-value) that yy can be at the peak of the circle?