8.2 Graphs of the other trigonometric functions  (Page 8/9)

Using the graphs of trigonometric functions to solve real-world problems

Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function .

Key equations

Key concepts

• The tangent function has period $\pi .$
• $f\left(x\right)=A\tan \left(Bx-C\right)+D$ is a tangent with vertical and/or horizontal stretch/compression and shift. See [link] , [link] , and [link] .
• The secant and cosecant are both periodic functions with a period of $2\pi .$ $f\left(x\right)=A\sec \left(Bx-C\right)+D$ gives a shifted, compressed, and/or stretched secant function graph. See [link] and [link] .
• $f\left(x\right)=A\csc \left(Bx-C\right)+D$ gives a shifted, compressed, and/or stretched cosecant function graph. See [link] and [link] .
• The cotangent function has period $\pi$ and vertical asymptotes at $0,\pm \pi ,\pm 2\pi ,\mathrm{...}.$
• The range of cotangent is $\left(-\infty ,\infty \right),$ and the function is decreasing at each point in its range.
• The cotangent is zero at $\pm \frac{\pi }{2},\pm \frac{3\pi }{2},\mathrm{...}.$
• $f\left(x\right)=A\cot \left(Bx-C\right)+D$ is a cotangent with vertical and/or horizontal stretch/compression and shift. See [link] and [link] .
• Real-world scenarios can be solved using graphs of trigonometric functions. See [link] .

Section exercises

Verbal