Graphs of the Other Trigonometric Functions: Apply It 1
Graph transformations of y=tan x and y=cot x.
Determine a function formula from a tangent or cotangent graph.
Graph transformations of y=sec x and y=csc x.
Determine a function formula from a secant or cosecant graph.
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.
Suppose the function [latex]y=5\tan\left(\frac{\pi}{4}t\right)[/latex] marks the distance in the movement of a light beam from the top of a police car across a wall where t is the time in seconds and y is the distance in feet from a point on the wall directly across from the police car.
Find and interpret the stretching factor and period.
Graph on the interval [0, 5].
Evaluate f(1) and discuss the function’s value at that input.
We know from the general form of [latex]y=A\tan(Bt)\\[/latex] that |A| is the stretching factor and π B is the period.
We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.
The period is [latex]\frac{\pi}{\frac{\pi}{4}}=\frac{\pi}{1}\times \frac{4}{\pi}=4[/latex]. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.
To graph the function, we draw an asymptote at [latex]t=2[/latex] and use the stretching factor and period.
period: [latex]f(1)=5\tan \left(\frac{\pi}{4}\left(1\right)\right)=5\left(1\right)=5[/latex]; after 1 second, the beam of has moved 5 ft from the spot across from the police car.
While sine and cosine create smooth wave patterns, the graphs of tangent, cotangent, secant, and cosecant have unique characteristics including vertical asymptotes and periodic gaps. These functions model real-world phenomena such as rotating light beams, shadows cast by the sun, and signal strength in telecommunications. In this page, you’ll apply these functions to analyze a lighthouse beacon and a solar panel tracking system.
A periodic signal has the following characteristics:
Period: 8 seconds
Minimum absolute value: 5
Maximum absolute value: approaches infinity
Vertical asymptotes at [latex]t = 2, 6, 10, ...[/latex]
Write a function in the form [latex]y = A\sec(Bt)[/latex].
Step 1: Find [latex]B[/latex] using the period. [latex]\begin{aligned} \frac{2\pi}{B} &= 8 \ B &= \frac{2\pi}{8} = \frac{\pi}{4} \end{aligned}[/latex] Step 2: Verify asymptote location. Asymptotes occur when [latex]\frac{\pi}{4}t = \frac{\pi}{2} + n\pi[/latex] [latex]t = 2 + 4n[/latex], giving [latex]t = 2, 6, 10, ...[/latex] ✓ Step 3: Find [latex]A[/latex]. The minimum absolute value equals [latex]|A|[/latex], so [latex]|A| = 5[/latex]. We’ll use [latex]A = 5[/latex] (positive). Function: [latex]y = 5\sec\left(\frac{\pi}{4}t\right)[/latex]
The height (in feet) a crane cable makes with the ground is modeled by [latex]h = 50\cot\left(\frac{\pi}{8}t\right)[/latex] where [latex]t[/latex] is time in seconds. Find the period and evaluate [latex]h(2)[/latex].
