🔺Trigonometry Unit 4 – Graphs of Sine and Cosine Functions

Key Concepts

• Sine and cosine are periodic functions that repeat their values at regular intervals
• Amplitude measures the height of the wave from the midline to the peak (or trough)
• Period is the length of one complete cycle of the function
  • Calculated using the formula $\frac{2\pi}{b}$, where $b$ is the absolute value of the coefficient of $x$
• Phase shift moves the graph horizontally to the left or right
  • Determined by the constant $c$ in the argument of the function $\sin(b(x-c))$ or $\cos(b(x-c))$
• Vertical shift moves the graph up or down
  • Determined by the constant $d$ in the function $a\sin(b(x-c))+d$ or $a\cos(b(x-c))+d$
• Transformations of sine and cosine graphs involve changes in amplitude, period, phase shift, and vertical shift
• Real-world applications of sine and cosine functions include modeling periodic phenomena (sound waves, tides, seasons)

Sine and Cosine Functions Basics

• Sine and cosine are trigonometric functions that relate angles to the lengths of the sides of a right triangle
• In the unit circle, sine represents the y-coordinate, and cosine represents the x-coordinate of a point on the circle
• Sine and cosine functions have a domain of all real numbers and a range of [-1, 1]
• The sine function is defined as $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• The cosine function is defined as $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Sine and cosine are odd and even functions, respectively
  • $\sin(-\theta) = -\sin(\theta)$ (odd function)
  • $\cos(-\theta) = \cos(\theta)$ (even function)

Graphing Sine and Cosine

• The basic sine function is $y = \sin(x)$, where $x$ is in radians
• The basic cosine function is $y = \cos(x)$, where $x$ is in radians
• Both sine and cosine graphs have a period of $2\pi$ and an amplitude of 1
• The sine graph starts at the origin (0, 0) and has a range of [-1, 1]
• The cosine graph starts at (0, 1) and has a range of [-1, 1]
• To graph sine and cosine functions, plot key points at multiples of $\frac{\pi}{2}$ and connect them smoothly
• The sine graph is symmetric about the origin, while the cosine graph is symmetric about the y-axis

Amplitude and Period

• Amplitude ($a$) is the distance from the midline to the maximum or minimum point of the graph
  • Determined by the coefficient $a$ in the function $a\sin(x)$ or $a\cos(x)$
  • If $|a| > 1$, the graph is stretched vertically; if $0 < |a| < 1$, the graph is compressed vertically
• Period ($p$) is the length of one complete cycle of the function
  • Calculated using the formula $p = \frac{2\pi}{|b|}$, where $b$ is the coefficient of $x$ inside the function
  • If $|b| > 1$, the period decreases, and the graph is compressed horizontally; if $0 < |b| < 1$, the period increases, and the graph is stretched horizontally

Phase Shifts and Vertical Shifts

• Phase shift ($c$) moves the graph horizontally to the left or right
  • Determined by the constant $c$ in the argument of the function $\sin(x-c)$ or $\cos(x-c)$
  • If $c > 0$, the graph shifts to the right; if $c < 0$, the graph shifts to the left
  • The phase shift is calculated by $\frac{c}{b}$, where $b$ is the coefficient of $x$ inside the function
• Vertical shift ($d$) moves the graph up or down
  • Determined by the constant $d$ in the function $\sin(x)+d$ or $\cos(x)+d$
  • If $d > 0$, the graph shifts up; if $d < 0$, the graph shifts down
• The general forms of sine and cosine functions with phase and vertical shifts are:
  • $y = a\sin(b(x-c))+d$
  • $y = a\cos(b(x-c))+d$

Transformations of Sine and Cosine Graphs

• Transformations of sine and cosine graphs involve changes in amplitude, period, phase shift, and vertical shift
• To graph a transformed sine or cosine function:
  1. Determine the amplitude $|a|$ and reflect the graph across the x-axis if $a < 0$
  2. Calculate the period $\frac{2\pi}{|b|}$ and adjust the horizontal scale accordingly
  3. Find the phase shift $\frac{c}{b}$ and shift the graph left or right
  4. Identify the vertical shift $d$ and move the graph up or down
• The order of transformations matters: amplitude and reflection, period, phase shift, and vertical shift
• When given an equation in the form $y = a\sin(b(x-c))+d$ or $y = a\cos(b(x-c))+d$, identify the values of $a$, $b$, $c$, and $d$ to determine the transformations

Real-World Applications

• Sine and cosine functions can model various periodic phenomena in the real world
• Sound waves: The pressure variation in sound waves can be modeled using sine or cosine functions
  • The amplitude represents the loudness, and the period represents the pitch
• Tides: The rise and fall of ocean tides can be approximated using sine or cosine functions
  • The amplitude represents the difference between high and low tides, and the period is about 12 hours and 25 minutes
• Seasons: The variation in daylight hours throughout the year can be modeled using a sine function
  • The amplitude represents the difference between the longest and shortest days, and the period is one year
• Electrical currents: Alternating current (AC) can be represented by a sine function
  • The amplitude represents the peak voltage, and the period is determined by the frequency of the current

Practice Problems and Tips

• Practice graphing sine and cosine functions with various transformations
  • Identify the amplitude, period, phase shift, and vertical shift
  • Sketch the graph by applying the transformations in the correct order
• Determine the equation of a sine or cosine function given its graph
  • Identify the amplitude, period, phase shift, and vertical shift from the graph
  • Write the equation in the general form $y = a\sin(b(x-c))+d$ or $y = a\cos(b(x-c))+d$
• Solve problems involving real-world applications of sine and cosine functions
  • Identify the periodic phenomenon and determine the appropriate function (sine or cosine)
  • Interpret the amplitude and period in the context of the problem
• Remember the key formulas and properties of sine and cosine functions
  • Period: $\frac{2\pi}{|b|}$
  • Phase shift: $\frac{c}{b}$
  • Odd function: $\sin(-\theta) = -\sin(\theta)$
  • Even function: $\cos(-\theta) = \cos(\theta)$