Comprehensive Notes on Periodic Functions and Trigonometric Identities

Foundations of Periodic and Trigonometric Functions

• Definition of a Periodic Function: A function is classified as periodic if it exhibits a repeating pattern of $y$ values at regular intervals. This repetition suggests that the function's behavior can be predicted over time or distance based on its past cycles.

• The Cycle: A cycle represents the smallest unique portion of the graph that repeats. Identifying one cycle is necessary for determining the properties of the entire function.

• The Period: The period is the horizontal length (measured along the x-axis) of one complete cycle.

  • Determining Period Algebraically: To find the period, identify the x-coordinate at the start of a cycle ($x_1$) and the x-coordinate at the start of the subsequent cycle ($x_2$). The period is calculated as:     $P = x_2 - x_1$

  • Example Calculation: If a cycle starts at $x = -4$ and the next starts at $x = 1$, the period is calculated as $1 - (-4) = 5\,units$. If a graph starts at $x = 0$ and ends a cycle at $x = 7$, the period is $7 - 0 = 7$.

• Evaluating Periodic Functions: To find local values of a function at large domain points, the period can be subtracted repeatedly.

  • Example: If the period is $7$ and you need to find $f(20)$, you can calculate $f(20 - 7) = f(13)$. If $f(13)$ is still outside the known graph range, subtract the period again: $f(13 - 7) = f(6)$. If $f(6) = -1$, then $f(20) = -1$.

Amplitude and the Axis of the Curve

• The Maximum and Minimum: The maximum value is the highest peak reached by the function, and the minimum is the lowest trough.

• Amplitude: This is the distance from the horizontal midline (axis of the curve) to either the maximum or the minimum. It represents half of the total vertical height of the function.

  • Formula: $\text{Amplitude} = \frac{\text{max} - \text{min}}{2}$

  • Example: For a function with a maximum of $3$ and a minimum of $-1$, the calculation is:     $\frac{3 - (-1)}{2} = \frac{4}{2} = 2\,units$

• Axis of the Curve: The horizontal line that is the average of the maximum and minimum values. It is expressed by the equation:     $y = \frac{\text{max} + \text{min}}{2}$

Properties of Basic Trigonometric Functions

• The Sine Function ($y = \sin(\theta)$):

  • Initial Values: $0^\circ = 0$, $30^\circ = 0.5$, $90^\circ = 1$, $180^\circ = 0$, $270^\circ = -1$, $360^\circ = 0$.

  • Amplitude: $1$

  • Period: $360^\circ$

  • Axis of the Curve: $y = 0$

  • Domain: $\theta \in \mathbb{R}$, often specific cycles are observed (e.g., $-360^\circ \leq \theta \leq 720^\circ$).

  • Range: $\{y \in \mathbb{R} | -1 \leq y \leq 1\}$

• The Cosine Function ($y = \cos(\theta)$):

  • Initial Values: $0^\circ = 1$, $60^\circ = 0.5$, $90^\circ = 0$, $180^\circ = -1$, $270^\circ = 0$, $360^\circ = 1$.

  • Amplitude: $1$

  • Period: $360^\circ$

  • Axis of the Curve: $y = 0$

  • Domain: $\theta \in \mathbb{R}$

  • Range: $\{y \in \mathbb{R} | -1 \leq y \leq 1\}$

Radian and Degree Measure

• Definition: Degrees and Radians are units for measuring angles.

• The Relationship: A full rotation in a circle is $360^\circ$ or $2\pi\,radians$. This simplifies to the fundamental conversion identity:     $\pi\,radians = 180^\circ$

• Conversion Formulas:

  • Degrees to Radians: Multiply by $\frac{\pi}{180^\circ}$.

  • Radians to Degrees: Multiply by $\frac{180^\circ}{\pi}$.

• Worked Examples:

  • Convert $60^\circ$ to radians: $60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}\,rad \approx 1.046\,rad$

  • Convert $45^\circ$ to radians: $45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}\,rad \approx 0.785\,rad$

  • Convert $\frac{\pi}{2}\,radians$ to degrees: $\frac{\pi}{2} \times \frac{180^\circ}{\pi} = 90^\circ$

Transformations of Periodic Functions

All algebraic transformation principles apply to trigonometric functions using the standard model: $y = a f(k(x - d)) + c$

• Vertical Stretches and Compressions ($a$):

  • If |a| > 1, the function undergoes a vertical expansion.

  • If 0 < |a| < 1, the function undergoes a vertical compression.

  • The value of $|a|$ always represents the amplitude of the function.

  • Example: For $y = 4\cos(x)$, the amplitude is $4$. The range becomes $\{y \in \mathbb{R} | -4 \leq y \leq 4\}$.

• Horizontal Stretches and Compressions ($k$):

  • If k > 1, the function undergoes a horizontal compression.

  • If 0 < k < 1, the function undergoes a horizontal expansion.

  • The period is directly affected by $k$:     $\text{Period} = \frac{360^\circ}{k}\,\text{or}\,\text{Period} = \frac{2\pi}{k}$

  • Example: For $y = \sin(3x)$, the period is $\frac{360^\circ}{3} = 120^\circ$.

• Horizontal Translations (Phase Shifts, $d$):

  • A shift to the right occurs if the term is units of $(x - d)$.

  • A shift to the left occurs if the term is units of $(x + d)$.

  • Example: For $y = 0.5\sin(x + 90^\circ)$, the graph is shifted $90^\circ$ to the left.

• Vertical Translations ($c$):

  • The value of $c$ shifts the entire function up or down. This value determines the location of the axis of the curve ($y = c$).

Real-World Modelling Applications

• Ferris Wheel Problem:

  • Data: Diameter = $38\,m$, rotation time = $6\,min$, boarding height = $4\,m$.

  • Analysis:

    • The radius is $19\,m$, so the amplitude $a = 19$.

    • The lowest point is $4\,m$, and the highest point is $4 + 38 = 42\,m$.

    • The axis of curve is $y = 23\,m$ ($\frac{4 + 42}{2} = 23$).

    • Period is $6\,min = 360\,s$.

  • Equation: Since the rider starts at the bottom, a negative cosine function is easiest to use:     $h(t) = -19\cos(t) + 23$ (if time is in degrees relative to the period).

  • Height Prediction: To find height at $1260\,seconds$, substitute into the function: $h(1260) = 42\,m$.

• Mass on a Spring:

  • Data: 12 cycles in 24 seconds, amplitude = $3.0\,cm$, minimum height = $4.0\,cm$, starts at $7.0\,cm$ moving up at $t=0$.

  • Analysis:

    • Period = $\frac{24\,s}{12\,cycles} = 2\,s/cycle$.

    • AOC = $4.0 + 3.0 = 7.0\,cm$.

    • Maximum height = $7.0 + 3.0 = 10.0\,cm$.

  • Equation: Because it starts at the axis ($y=7$) and moves upward, a sine function is appropriate:     $h(t) = 3\sin(180t) + 7$

Trigonometric Ratios and the CAST Rule

• Coordinates on a Circular Path: For any point $P(x, y)$ on a terminal arm of angle $\theta$:

  • $r = \sqrt{x^2 + y^2}$

  • $\sin(\theta) = \frac{y}{r}$

  • $\cos(\theta) = \frac{x}{r}$

  • $\tan(\theta) = \frac{y}{x}$

• The CAST Rule: Determines which primary ratios are positive in each quadrant:

  • Quadrant I (All): All ratios (Sine, Cosine, Tangent) are positive.

  • Quadrant II (Sine): Only Sine is positive.

  • Quadrant III (Tangent): Only Tangent is positive.

  • Quadrant IV (Cosine): Only Cosine is positive.

• Reciprocal Ratios:

  • Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{r}{y}$

  • Secant: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{r}{x}$

  • Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{x}{y}$

Special Triangles and Exact Values

Trigonometric ratios for specific angles should be expressed in exact radical form rather than decimals.

• 45-45-90 Triangle: Sides are $1, 1, \sqrt{2}$.

  • $\sin(45^\circ) = \frac{1}{\sqrt{2}}$

  • $\cos(45^\circ) = \frac{1}{\sqrt{2}}$

  • $\tan(45^\circ) = 1$

• 30-60-90 Triangle: Sides are $1, \sqrt{3}, 2$.

  • $\sin(30^\circ) = \frac{1}{2}$; $\sin(60^\circ) = \frac{\sqrt{3}}{2}$

  • $\cos(30^\circ) = \frac{\sqrt{3}}{2}$; $\cos(60^\circ) = \frac{1}{2}$

  • $\tan(30^\circ) = \frac{1}{\sqrt{3}}$; $\tan(60^\circ) = \sqrt{3}$

• Examples:

  • Find exact value of $\tan(150^\circ)$.: Angle is in Quadrant II, reference angle is $180^\circ - 150^\circ = 30^\circ$. Since Tangent is negative in QII: $\tan(150^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}}$.

  • Find value of $\sec(210^\circ)$.: Reference angle is $210^\circ - 180^\circ = 30^\circ$. Secant is the reciprocal of Cosine. In QIII, Cosine is negative. $\cos(210^\circ) = -\frac{\sqrt{3}}{2}$, therefore $\sec(210^\circ) = -\frac{2}{\sqrt{3}}$.