Example 10 from Algebra and Trigonometry, 8.2 Graphs of the Other Trigonometric Functions

Using Trigonometric Functions to Solve Real-World Scenarios

Suppose the function $y = 5 \tan (\frac{π}{4} t)$ 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.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ We know from the general form of $y = A \tan (B t)$ that $| A |$ is the stretching factor and $\frac{π}{B}$ is the period.
  
  Figure 16
  
  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 $\frac{π}{\frac{π}{4}} = \frac{π}{1} \cdot \frac{4}{π} = 4.$ 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.
2. ⓑ To graph the function, we draw an asymptote at $t = 2$ and use the stretching factor and period. See Figure 17
  
  Figure 17
3. ⓒ period: $f (1) = 5 \tan (\frac{π}{4} (1)) = 5 (1) = 5;$ after 1 second, the beam of has moved 5 ft from the spot across from the police car.

How did it go?

Solution

Solution