Chapter 6 Part 4: Real-World Applications of Trigonometry

Welcome back to Professor Baker's Math Class! In this session, we explored the practical applications of sinusoidal functions and right triangle trigonometry. Let's review the key concepts we covered:

  • Evaluating Sinusoidal Functions: Learn how to evaluate sinusoidal functions that model real-world scenarios. For example, consider the height $h$ of a mass on a spring moving with simple harmonic motion, given by the formula: $$h = a \cos(\omega t - c)$$ where $a$ is the amplitude, $\omega$ is the angular frequency, $t$ is the time, and $c$ is the phase shift. Given $a = 7.9$ m, $\omega = \frac{3\pi}{7}$ rad/s, $t = 5$ s, and $c = \frac{3\pi}{7}$ rad, we found $h = 4.93$ m.
  • Sinusoidal Modeling: Develop sinusoidal models for real-world situations such as the displacement $d$ of a buoy floating in the ocean. If the buoy's motion has amplitude 3 ft and period 8 seconds, and its displacement at $t=0$ is 3 ft (moving upward), the model is: $$d = 3\cos(\frac{\pi}{4}t)$$
  • Graphing Sinusoidal Functions: Learn to sketch sinusoidal functions that model real-world situations and use the graph to approximate solutions to equations. We examined scenarios where minimum and maximum values help define the curve, like one with a minimum at (0, 0.5) and a maximum at (1, 5.5), leading to a function of the form: $$f(t) = -2.5\cos(\pi t) + 3$$
  • Right Triangle Trigonometry:
    • Finding Lengths: Use trigonometric functions (sine, cosine, tangent) to find unknown lengths in word problems involving one or two right triangles. Remember SOH CAH TOA!
    • Angles of Elevation and Depression: Apply trigonometry to find angles of elevation or depression in real-world problems. For example, determining the angle of depression of a plane from the airport.
    • Distance, Rate, and Time: Combining trigonometric functions with the formula $d = rt$ in real-world situations. For example, if a plane descends at a 2-degree angle at 400 km/hr, you can use trigonometry to determine altitude loss over a certain time period.

Keep practicing these concepts, and you'll become a trigonometry master in no time! Remember to review your notes and work through additional examples. Don't hesitate to ask questions in class or during office hours. Keep up the great work!