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Trig Derivative Cycle

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About This MicroSim

This interactive visualization demonstrates the cyclic nature of trigonometric derivatives. One of the beautiful patterns in calculus is that taking the derivative of sine and cosine functions creates a repeating cycle:

\[\frac{d}{dx}\sin(x) = \cos(x) \rightarrow \frac{d}{dx}\cos(x) = -\sin(x) \rightarrow \frac{d}{dx}(-\sin(x)) = -\cos(x) \rightarrow \frac{d}{dx}(-\cos(x)) = \sin(x)\]

After four derivatives, you return to the original function!

How to Use

  1. Take Derivative Button: Click to advance to the next function in the cycle
  2. Reset Button: Return to sin(x) as the starting function
  3. Show Both Graphs Checkbox: Toggle to see both f(x) and f'(x) graphs simultaneously

The Derivative Cycle

The circular diagram on the left shows the four-step cycle:

Step Function Derivative
1 sin(x) cos(x)
2 cos(x) -sin(x)
3 -sin(x) -cos(x)
4 -cos(x) sin(x)

The color coding in the circular diagram matches the graphs, helping you visually connect each function to its derivative.

Iframe Embedding

Place the following line in your website to include this MicroSim in your course:

1
<iframe src="https://dmccreary.github.io/calculus/sims/trig-derivative-cycle/main.html" height="482px" scrolling="no"></iframe>

Lesson Plan

Learning Objective

Students will be able to recall the cyclic pattern of trigonometric derivatives (sin, cos, -sin, -cos) and identify the derivative of any function in the cycle.

Grade Level

High School (Grades 11-12), AP Calculus

Duration

10-15 minutes

Prerequisites

  • Understanding of what a derivative represents
  • Familiarity with sine and cosine functions
  • Basic graphing skills

Warm-Up Activity (3 minutes)

Ask students: "What do you notice about the graphs of sin(x) and cos(x)? How are they related?"

Main Activity (7-10 minutes)

  1. Exploration Phase: Have students click through all four derivatives once while observing:
  2. How the graphs change
  3. The pattern in the circular diagram

  4. What happens after 4 clicks

  5. Pattern Recognition: Ask students to predict:

  6. What is the 5th derivative of sin(x)?
  7. What is the 8th derivative of sin(x)?

  8. What is the 100th derivative of sin(x)?

  9. Discussion Questions:

  10. Why does this cycle happen? (Hint: Think about phase shifts)
  11. How could you use this pattern to quickly find high-order derivatives?

Assessment Questions

  1. What is the derivative of cos(x)?
  2. If you take the derivative of sin(x) three times, what do you get?
  3. Why does the cycle repeat after exactly 4 derivatives?

Extension Activities

  • Have students explore what happens with sin(2x) or cos(3x)
  • Connect to the unit circle and phase shifts
  • Explore the relationship to Euler's formula: \(e^{ix} = \cos(x) + i\sin(x)\)

References