Trig Function Graphs

Welcome to this section dedicated to mastering trigonometric functions and their transformations. Our objective is to help you gain proficiency in graphing, transforming functions, and solving problems efficiently, thereby enhancing your exam performance.

Introduction

Trigonometric functions play a crucial role in mathematics, with practical applications in various fields such as sound wave analysis. Let's delve into sine, cosine, and tangent functions to understand their graphs.

Key Concepts

Graph Recognition

• Standard Shapes:

  • Sine and Cosine:
    • Exhibit wavelike patterns.
    • Noticeable peaks and troughs.
  • Tangent:
    • Features vertical asymptotes.
    • Displays recurring undefined values.

• Periodicity:

  • Sine and Cosine: Repeat every 360° or $2\pi$.
  • Tangent: Repeats every 180° or $\pi$.

• Amplitude:

  • Sine and Cosine: Vary between -1 and 1.
  • Tangent: Has no finite amplitude due to asymptotes.

Graph Sketching Guidelines

• Key Points Identification:
  • Highlight maxima, minima, and zeros using diagrams.
  • Example of sketching:
  \sin(0°) &= 0, \\ \sin(90°) &= 1, \\ \sin(180°) &= 0, \\ \sin(270°) &= -1, \\ \sin(360°) &= 0. \end{align*}$$
• Behaviour in Degrees and Radians:
  • Convert degrees to radians: $180° = \pi$ radians.

Graph Sketch Examples

• Review the sine function and cosine function across one period.

Define and Relate Reciprocal Functions

• Cosecant ($y = \csc x = \frac{1}{\sin x}$): Closely linked to the sine function.
• Secant ($y = \sec x = \frac{1}{\cos x}$): Directly associated with the cosine function.
• Cotangent ($y = \cot x = \frac{1}{\tan x}$): Directly related to the tangent function.
• Highlight their reciprocal nature for a robust understanding.

Graph Characteristics

• Periodicity:
  • Inherited from their primary counterparts: sine, cosine, and tangent.
• Vertical Asymptotes:
  • Occur where the primary functions equal zero.
• Intersections and Unique Behaviours:
  • Examine points where intersections occur.

Detailed Sketching Guidelines

Graph Sketching

• Step 1: Start with sketching the primary functions: $y = \sin x$, $y = \cos x$, $y = \tan x$.

• Step 2: Overlay these with their reciprocal functions: $y = \csc x$, $y = \sec x$, $y = \cot x$.

• Labelling Features:

  • Clearly denote vertical asymptotes and critical points.

Graph Examples

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Transformations of Trigonometric Functions

Transformation Formula

• Formula: The transformation formula for trigonometric functions is represented as: $y = kf(a(x + b)) + c$

• Explanations:

  • Amplitude (k): Modifies the wave's height.
  • Frequency (a): Adjusts the number of cycles within a given interval.
  • Horizontal Shift (b): Shifts the graph laterally.
  • Vertical Shift (c): Moves the graph vertically.

• Impact:

Parameter	Effect on Graph	Example (y = k \sin(a(x + b)) + c)
k	Modifies amplitude	e.g., $y = 2\sin(x)$ doubles the amplitude
a	Alters frequency & period	e.g., $y = \sin(2x)$ halves the period
b	Horizontal shift	e.g., $y = \sin(x - \pi/4)$ shifts right
c	Vertical shift	e.g., $y = \sin(x) + 3$ shifts upward

Graphical Examples

• Example 1: Illustration of transformations on the sine function.
• Example 2: Depiction of phase shift in the cosine function.

Worked Example: Transformation of Sine Function

Let's analyse the function $y = 3\sin(2(x-\pi/4))+1$:

1. Amplitude: $k = 3$, so the amplitude is 3 (range is from -3 to 3 before vertical shift)
2. Frequency: $a = 2$, so the period is $\pi$ (half the original period)
3. Horizontal shift: $b = \pi/4$, shifting the graph $\pi/4$ units to the right
4. Vertical shift: $c = 1$, raising the graph by 1 unit

The resulting graph will oscillate between -2 and 4, completing a full cycle every $\pi$ units, and starting its cycle $\pi/4$ units to the right of the origin.

Misconceptions and Errors

• Common Issues: Address misunderstandings regarding phase or amplitude.

Comparative Analysis

Phase Difference

• Phase Shift: The sine function leads the cosine function by $\frac{\pi}{2}$ radians.

  

Symmetry

• Sine: An odd function, symmetric about the origin.

• Cosine: An even function, symmetric about the y-axis.

• Tangent: Also classified as an odd function.

  

Example Questions with Solutions

1. Question: Sketch the graph of $y = 2\cos(3x) - 1$ for $0 \leq x \leq 2\pi$.

   Solution:

   • Amplitude: $k = 2$ (graph oscillates between -2 and 2 before shifting)
   • Period: $\frac{2\pi}{3}$ (as $a = 3$)
   • Vertical shift: $c = -1$ (graph is shifted down by 1 unit)
   • The graph will oscillate between -3 and 1, completing 3 full cycles over the interval $[0,2\pi]$.

2. Question: Identify the amplitude, period, and phase shift of $y = -\sin(2x+\pi)+3$.

   Solution:

   • Rewriting as $y = -\sin(2(x+\pi/2))+3$
   • Amplitude: $k = 1$ (negative sign flips the graph)
   • Period: $\pi$ (as $a = 2$)
   • Horizontal shift: $b = -\pi/2$ (shifting left by $\pi/2$)
   • Vertical shift: $c = 3$ (shifting up by 3 units)