Pre-Calculus 12 Notes

By: Gen L.

In partnership with Hyperion University, 2024

Lesson 4: Applications

Congruent Equations

Write an equation that gives the same graph as .

  • There's an infinite amount similar to this one:
  • How about cosine?

Equation from Extrema Points

A cosine curve has a maximum at , with the closest minimum at . Find a possible equation.

  1. Determine .
  2. Since it's a cosine graph, we can use the maximum to determine

Equation from Extrema, cont.

, min: , max:

  1. Determine
    • After evaluating this, we get
  2. Determine
    • Since we have half of the period, we can take to get the whole period.
    • , then just take , which gives us

Applications of Trig Functions

The previous examples are cool, but what about actual applications?

  • Tidal heights,
  • Ferris wheels,
  • Springs,
  • Sunrise / sunsets,
  • Pendula (pendulums),
  • Heights of wheels, etc.

Tidal Example

High tide is at 6 hours, and the next low tide is at 12.4 hours.

  1. Determine an equation of this sinusoidal function
    • Use the skills from determining equations from extrema.
  2. At 10 hours, what is the depth?
  3. When is the first time the water is at ? [desmos]
  4. How long is the water above ? [desmos]

Tidal Answers

  3. There's a way to solve this algebraically, but

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