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Exercises Trigonometry

Example 8 from Precalculus, 7.6 Modeling with Trigonometric Functions

Finding the Displacement, Period, and Frequency, and Graphing a Function

For the given functions,

Find the maximum displacement of an object.
Find the period or the time required for one vibration.
Find the frequency.
Sketch the graph.

ⓐ $y = 5 \sin (3 t)$
ⓑ $y = 6 \cos (π t)$
ⓒ $y = 5 \cos (\frac{π}{2} t)$

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. ⓐ y=5sin(3t)
  1. The maximum displacement is equal to the amplitude, $| a |,$ which is 5.
  2. The period is $\frac{2 π}{ω} = \frac{2 π}{3} .$
  3. The frequency is given as $\frac{ω}{2 π} = \frac{3}{2 π} .$
  4. See Figure 12. The graph indicates the five key points.
    
    Figure 12
2. ⓑ y=6cos(πt)
  1. The maximum displacement is $6.$
  2. The period is $\frac{2 π}{ω} = \frac{2 π}{π} = 2.$
  3. The frequency is $\frac{ω}{2 π} = \frac{π}{2 π} = \frac{1}{2} .$
  4. See Figure 13.
    
    Figure 13
3. ⓒ y=5cos(π2)t
  1. The maximum displacement is $5.$
  2. The period is $\frac{2 π}{ω} = \frac{2 π}{\frac{π}{2}} = 4.$
  3. The frequency is $\frac{1}{4} .$
  4. See Figure 14.
    
    Figure 14

How did it go?

PreviousInterpreting the Periodic Behavior Equation Next Modeling Damped Harmonic Motion

From Precalculus (1st edition) by OpenStax (Rice University), section 7.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.