🔺Trigonometry Unit 5 – Graphs of Other Trigonometric Functions

Key Concepts and Definitions

• Trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant, which are defined in terms of angles and ratios of sides in a right triangle
• Period of a trigonometric function represents the length of one complete cycle on the graph, with sine and cosine having a period of $2\pi$, while tangent and cotangent have a period of $\pi$
• Amplitude of a trigonometric function measures the height of the graph from the midline to the maximum or minimum point, with the parent functions having an amplitude of 1
• Phase shift of a trigonometric function moves the graph horizontally, with a positive phase shift moving the graph to the left and a negative phase shift moving it to the right
• Vertical shift of a trigonometric function moves the graph up or down, with a positive vertical shift moving the graph up and a negative vertical shift moving it down
• Asymptotes are vertical lines that the graph of a function approaches but never touches, occurring in tangent, cotangent, secant, and cosecant functions
  • Tangent and cotangent functions have asymptotes at $x=\frac{\pi}{2}+\pi k$ and $x=\pi k$, respectively, where $k$ is an integer
  • Secant and cosecant functions have asymptotes at $x=\frac{\pi}{2}+\pi k$ and $x=\pi k$, respectively, where $k$ is an integer

Graphing Cosine Functions

• The cosine function, denoted as $\cos(x)$, is defined as the ratio of the adjacent side to the hypotenuse in a right triangle
• The parent function of cosine is $f(x)=\cos(x)$, which has a domain of all real numbers and a range of $[-1,1]$
• The graph of the cosine function is a smooth, continuous wave that oscillates between -1 and 1, with a period of $2\pi$
• The cosine function has maximum points at $x=2\pi k$ and minimum points at $x=\pi+2\pi k$, where $k$ is an integer
• The cosine function is an even function, meaning that $\cos(-x)=\cos(x)$ for all $x$ in the domain
• To graph a cosine function with a phase shift, vertical shift, or amplitude change, use the general form $f(x)=A\cos(B(x-C))+D$, where:
  • $A$ represents the amplitude (vertical stretch or compression)
  • $B$ represents the period (horizontal stretch or compression)
  • $C$ represents the phase shift (horizontal shift)
  • $D$ represents the vertical shift

Graphing Tangent Functions

• The tangent function, denoted as $\tan(x)$, is defined as the ratio of the opposite side to the adjacent side in a right triangle
• The parent function of tangent is $f(x)=\tan(x)$, which has a domain of all real numbers except $x=\frac{\pi}{2}+\pi k$, where $k$ is an integer, and a range of all real numbers
• The graph of the tangent function consists of repeated S-shaped curves with asymptotes at $x=\frac{\pi}{2}+\pi k$, where $k$ is an integer
• The tangent function has a period of $\pi$, meaning that the graph repeats itself every $\pi$ units
• The tangent function is an odd function, meaning that $\tan(-x)=-\tan(x)$ for all $x$ in the domain
• To graph a tangent function with a phase shift or vertical stretch, use the general form $f(x)=A\tan(B(x-C))$, where:
  • $A$ represents the vertical stretch
  • $B$ represents the period (horizontal stretch or compression)
  • $C$ represents the phase shift (horizontal shift)

Graphing Cotangent Functions

• The cotangent function, denoted as $\cot(x)$, is defined as the reciprocal of the tangent function, or the ratio of the adjacent side to the opposite side in a right triangle
• The parent function of cotangent is $f(x)=\cot(x)$, which has a domain of all real numbers except $x=\pi k$, where $k$ is an integer, and a range of all real numbers
• The graph of the cotangent function consists of repeated U-shaped curves with asymptotes at $x=\pi k$, where $k$ is an integer
• The cotangent function has a period of $\pi$, meaning that the graph repeats itself every $\pi$ units
• The cotangent function is an odd function, meaning that $\cot(-x)=-\cot(x)$ for all $x$ in the domain
• To graph a cotangent function with a phase shift or vertical stretch, use the general form $f(x)=A\cot(B(x-C))$, where:
  • $A$ represents the vertical stretch
  • $B$ represents the period (horizontal stretch or compression)
  • $C$ represents the phase shift (horizontal shift)

Graphing Secant and Cosecant Functions

• The secant function, denoted as $\sec(x)$, is defined as the reciprocal of the cosine function, or the ratio of the hypotenuse to the adjacent side in a right triangle
• The cosecant function, denoted as $\csc(x)$, is defined as the reciprocal of the sine function, or the ratio of the hypotenuse to the opposite side in a right triangle
• The parent function of secant is $f(x)=\sec(x)$, which has a domain of all real numbers except $x=\frac{\pi}{2}+\pi k$, where $k$ is an integer, and a range of $(-\infty,-1]\cup[1,\infty)$
• The parent function of cosecant is $f(x)=\csc(x)$, which has a domain of all real numbers except $x=\pi k$, where $k$ is an integer, and a range of $(-\infty,-1]\cup[1,\infty)$
• The graph of the secant function consists of repeated U-shaped curves with asymptotes at $x=\frac{\pi}{2}+\pi k$, where $k$ is an integer
• The graph of the cosecant function consists of repeated U-shaped curves with asymptotes at $x=\pi k$, where $k$ is an integer
• Both secant and cosecant functions have a period of $2\pi$, meaning that the graph repeats itself every $2\pi$ units
• To graph a secant or cosecant function with a phase shift or vertical stretch, use the general forms $f(x)=A\sec(B(x-C))$ or $f(x)=A\csc(B(x-C))$, where:
  • $A$ represents the vertical stretch
  • $B$ represents the period (horizontal stretch or compression)
  • $C$ represents the phase shift (horizontal shift)

Transformations of Trigonometric Graphs

• Transformations of trigonometric graphs include changes in amplitude, period, phase shift, and vertical shift
• Amplitude changes affect the height of the graph, with a vertical stretch occurring when $|A|>1$ and a vertical compression occurring when $0<|A|<1$
• Period changes affect the length of one complete cycle, with a horizontal compression occurring when $|B|>1$ and a horizontal stretch occurring when $0<|B|<1$
• Phase shifts move the graph horizontally, with a positive phase shift ($C>0$) moving the graph to the left and a negative phase shift ($C<0$) moving the graph to the right
• Vertical shifts move the graph up or down, with a positive vertical shift ($D>0$) moving the graph up and a negative vertical shift ($D<0$) moving the graph down
• To apply multiple transformations, follow the order: amplitude change, period change, phase shift, and then vertical shift
• When given an equation in the form $f(x)=A\sin(B(x-C))+D$ or $f(x)=A\cos(B(x-C))+D$, identify the values of $A$, $B$, $C$, and $D$ to determine the transformations applied to the parent function

Applications and Real-World Examples

• Trigonometric functions have numerous applications in various fields, such as physics, engineering, and music
• In physics, trigonometric functions are used to model periodic phenomena, such as waves, oscillations, and rotations
  • Example: The motion of a pendulum can be modeled using the sine function, with the angle of the pendulum varying sinusoidally over time
• In engineering, trigonometric functions are used to analyze and design structures, such as bridges and buildings, by calculating angles, distances, and forces
  • Example: The height of a suspension bridge's cable at any point can be modeled using a cosine function, with the lowest point of the cable at the center of the bridge
• In music, trigonometric functions are used to describe the harmonics and waveforms of musical notes and instruments
  • Example: The sound waves produced by a guitar string can be modeled using a combination of sine functions with different frequencies and amplitudes
• Trigonometric functions are also used in navigation and GPS systems to calculate distances and angles between locations on the Earth's surface
  • Example: The great-circle distance between two points on the Earth's surface can be calculated using the haversine formula, which involves the sine function

Common Mistakes and How to Avoid Them

• Confusing the period of tangent and cotangent functions ($\pi$) with the period of sine, cosine, secant, and cosecant functions ($2\pi$)
  • Remember that tangent and cotangent functions have asymptotes that occur twice as frequently as those of secant and cosecant functions
• Incorrectly applying transformations to trigonometric functions, especially when multiple transformations are involved
  • Follow the order of transformations: amplitude change, period change, phase shift, and then vertical shift
• Forgetting to consider the domain restrictions when graphing tangent, cotangent, secant, and cosecant functions
  • Identify the asymptotes and exclude them from the domain
• Misinterpreting the signs of the phase shift and vertical shift in the general form of the equation
  • A positive phase shift ($C>0$) moves the graph to the left, while a negative phase shift ($C<0$) moves the graph to the right
  • A positive vertical shift ($D>0$) moves the graph up, while a negative vertical shift ($D<0$) moves the graph down
• Neglecting to consider the amplitude and period changes when sketching the graph of a transformed trigonometric function
  • Pay attention to the coefficients $A$ and $B$ in the general form of the equation to determine the amplitude and period changes
• Incorrectly identifying the period of a transformed trigonometric function
  • The period of a transformed function is given by $\frac{2\pi}{|B|}$, where $B$ is the coefficient of the input angle in the general form of the equation