TR-18: Graphing Sine and Cosine (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial will guide you through the process of graphing sine and cosine functions, using key concepts from trigonometry. Understanding how to graph these functions is essential for students in algebra and trigonometry, especially for those preparing for various international exams. By the end of this guide, you will be able to create accurate sine and cosine graphs and recognize their distinct characteristics.

Step 1: Understanding the Independent Variable

• Introduce theta (θ) as the independent variable.
• Recognize that theta represents the angle in radians or degrees and is typically plotted on the x-axis.
• Familiarize yourself with the unit circle, as it helps in visualizing the sine and cosine values for different angles.

Step 2: Recognizing the Basic Properties of Sine and Cosine

• Identify the amplitude, period, and phase shift of sine and cosine waves.
  • Amplitude: The maximum height of the wave from the center line, which is 1 for both sine and cosine functions.
  • Period: The length of one complete cycle of the wave.
    • For sine and cosine, the standard period is (2\pi) (approximately 6.28).
• Note the key points you will plot for one full cycle:
  • For sine: (0, 0), ((\frac{\pi}{2}), 1), ((\pi), 0), ((\frac{3\pi}{2}), -1), (2\pi, 0)
  • For cosine: (0, 1), ((\frac{\pi}{2}), 0), ((\pi), -1), ((\frac{3\pi}{2}), 0), (2\pi, 1)

Step 3: Graphing the Sine Function

• Plot the key points on a graph:
  1. Start at (0, 0).
  2. Move to ((\frac{\pi}{2}), 1).
  3. Continue to ((\pi), 0).
  4. Then plot ((\frac{3\pi}{2}), -1).
  5. Complete the cycle at (2\pi, 0).
• Connect these points smoothly to form the sine wave.

Step 4: Graphing the Cosine Function

• Plot the key points for the cosine function:
  1. Start at (0, 1).
  2. Move to ((\frac{\pi}{2}), 0).
  3. Continue to ((\pi), -1).
  4. Then plot ((\frac{3\pi}{2}), 0).
  5. Complete the cycle at (2\pi, 1).
• Connect these points smoothly to form the cosine wave.

Step 5: Distinguishing Between Sine and Cosine

• Use a mnemonic to remember the differences between sine and cosine:
  • Sine starts at zero and goes up, while cosine starts at one and goes down.
• Recognize the phase shift: the sine wave is a horizontal shift of the cosine wave to the right by (\frac{\pi}{2}).

Practical Tips

• Use graph paper or digital graphing tools for accuracy.
• Label your axes and key points clearly.
• Practice with different amplitudes and periods to see how they affect the graph.

Common Pitfalls to Avoid

• Mislabeling the axes can lead to incorrect interpretations of the graphs.
• Forgetting to mark the key points may result in a distorted graph.
• Confusing the sine and cosine functions, especially regarding their starting points.

Conclusion

Graphing sine and cosine functions is a fundamental skill in trigonometry. By following these steps, you can accurately create and interpret these graphs. As next steps, practice by graphing variations of these functions with different amplitudes and periods, and explore their applications in real-world contexts such as sound waves and oscillations.