MAC 1114 Module 2: Trigonometric Functions — Unit Circle, Functions, Periodicity, and Even/Odd Properties

The Unit Circle

• Definition: The Unit Circle is the circle of radius 1, centered at the origin in the coordinate plane. Its equation is $x^2 + y^2 = 1.$
• Terminology: Let \theta\ be a central angle in standard position. The point of intersection of the unit circle with the terminal side of \theta\ is called the Terminal Point of \theta.
• Right-triangle interpretation: Dropping a vertical line from the terminal point P(x, y) forms a right triangle with the initial side and the terminal side of \theta. The base length equals the x-coordinate, the height equals the y-coordinate. These relations help compute terminal point coordinates.
• Coordinates on the unit circle: On the unit circle, the coordinates of the terminal point are (cos(\theta), sin(\theta)). Thus x = cos(\theta), y = sin(\theta).
• Special key points on the unit circle (quadrantal angles):
  • 0° (0 rad): (1, 0)
  • 90° (\frac{\pi}{2} rad): (0, 1)
  • 180° (\pi rad): (-1, 0)
  • 270° (\frac{3\pi}{2} rad): (0, -1)
• Special angles in the first quadrant:
  • 45° = \frac{\pi}{4}: \left(\frac{\sqrt{2}}{2}, \\frac{\sqrt{2}}{2}\right)
  • 30° = \frac{\pi}{6}: \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)
  • 60° = \frac{\pi}{3}: \left(\frac{1}{2}, \\frac{\sqrt{3}}{2}\right)
• Derivation notes for special angles:
  • 45°: In a right triangle with a 45° angle, the triangle is isosceles; to fit the unit circle, choose the hypotenuse = 1, which yields legs = \\frac{\sqrt{2}}{2}.
  • 60°: Start with an equilateral triangle, each angle 60°. If the equilateral triangle has side length 2\alpha, halving yields a right triangle with short leg = \alpha, long leg = \alpha\sqrt{3}, and hypotenuse = 2\alpha. Normalizing to hypotenuse 1 gives \alpha = \frac{1}{2}, so the legs are \frac{1}{2} and \frac{\sqrt{3}}{2}, corresponding to (cos 60°, sin 60°) = (\frac{1}{2}, \frac{\sqrt{3}}{2}).
• Reflection to other quadrants:
  • 135° = \frac{3\pi}{4}: ( -\frac{\sqrt{2}}{2}, \\frac{\sqrt{2}}{2} )
  • 225° = \frac{5\pi}{4}: ( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} )
  • 315° = \frac{7\pi}{4}: ( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} )
• In general, the terminal point for angle \theta on the unit circle is (cos(\theta), sin(\theta)).

Trigonometric Functions: Definition from Terminal Point

• Six functions defined by the terminal point (x, y) of angle \theta:
  • \sin(\theta) = y
  • \cos(\theta) = x
  • \tan(\theta) = \dfrac{y}{x}
  • \csc(\theta) = \dfrac{1}{y}
  • \sec(\theta) = \dfrac{1}{x}
  • \cot(\theta) = \dfrac{x}{y}
• These definitions tie directly to the unit circle coordinates: x = cos(\theta), y = sin(\theta).
• Important domain note: \tan(\theta) is undefined when x = 0 (i.e., when cos(\theta) = 0).

Domain and Range

• Sine and Cosine:
  • Domain: \(\mathbb{R}\)
  • Range: \[ -1, 1 \
    \]
• Tangent:
  • Definition: \tan(\theta) = \dfrac{y}{x}
  • Undefined when cos(\theta) = 0 (x = 0)
  • Domain: \{\theta \in \mathbb{R} \mid \cos(\theta) \\neq 0\}\
  • In [0, 2\pi], the domain excludes \(\frac{\pi}{2}, \frac{3\pi}{2}\)
  • Range: \mathbb{R}
• Cosecant, Secant, Cotangent:
  • \csc(\theta) = \dfrac{1}{\sin(\theta)}: defined when \sin(\theta) \neq 0 \Rightarrow \theta \neq k\pi
  • \sec(\theta) = \dfrac{1}{\cos(\theta)}: defined when \cos(\theta) \neq 0 \Rightarrow \theta \neq \frac{\pi}{2} + k\pi
  • \cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}: defined when \sin(\theta) \neq 0 \Rightarrow \theta \neq k\pi
• Quick interval form in [0, 2\pi] for tan:
  • D(tan) = (0,\frac{\pi}{2})
    \cup (\frac{\pi}{2},\pi)
    \cup (\pi,\frac{3\pi}{2})
    \cup (\frac{3\pi}{2},2\pi)

Periodic Functions

• Definition: A function f is periodic if there exists a positive number c such that for all real \theta, f(\theta + c) = f(\theta).
• The smallest such c is the period of f.
• Sin and Cosine:
  • \sin(\theta + 2\pi) = \sin(\theta)
  • \cos(\theta + 2\pi) = \cos(\theta)
  • These are minimal periods for sin and cos (there is no smaller positive period that works for all \theta).
• Tangent:
  • Tangent has period \pi: \tan(\theta + \pi) = \tan(\theta).
• Conceptual note: coterminal angles share the same terminal point on the unit circle, which underlies the periodic repetition of sine and cosine values.
• Exercises (sample results):
  • For \theta = \frac{25\pi}{4}: \sin(\theta) = \dfrac{\sqrt{2}}{2}, \cos(\theta) = \dfrac{\sqrt{2}}{2}.
  • For \theta = 780\circ: 780° \equiv 60° \Rightarrow \sin(780\circ) = \dfrac{\sqrt{3}}{2}, \cos(780\circ) = \dfrac{1}{2}.
  • For \theta = -570\circ: \,-570° \equiv 150° \Rightarrow \sin(-570\circ) = \dfrac{1}{2}, \cos(-570\circ) = -\dfrac{\sqrt{3}}{2}.

Even and Odd Properties

• Definitions:
  • A function f is even if \forall x, f(-x) = f(x).
  • A function f is odd if \forall x, f(-x) = -f(x).
• Geometric interpretation on the unit circle:
  • If the terminal point for angle \theta is (x, y), then the terminal point for -\theta is (x, -y) (reflection across the x-axis).
  • Therefore, cos( -\theta ) = cos(\theta) (cos is even) and sin( -\theta ) = - sin(\theta) (sin is odd).
• Consequences for trigonometric functions:
  • \cos(-\theta) = \cos(\theta)
• \sin(-\theta) = -\sin(\theta).
• Quick checks:
  • Example: \sin(-\frac{\pi}{6}) = -\frac{1}{2}, \cos(-\frac{\pi}{6}) = \frac{\sqrt{3}}{2}.
  • If \cos(\theta) = -\frac{1}{\sqrt{5}}, then \cos(-\theta) = -\frac{1}{\sqrt{5}}.
  • If \sin(\theta) = \frac{3}{5}, then \sin(-\theta) = -\frac{3}{5}.
• Parity rules for sums, differences, and products (when f is even and g is odd):
  • (f + g) is in general neither even nor odd.
  • (f - g) is in general neither even nor odd.
  • (f \cdot g) is odd (even × odd = odd).
• Practical note: Even/odd properties extend to angles in all quadrants; reflecting across the x-axis flips the sign of the sine component while leaving cosine unchanged.

References

• The material cites multiple online resources for unit circle visuals, periodicity, and even/odd properties.