4.1 Graphing Sine and Cosine Functions

Sine and cosine functions are the backbone of trigonometry. They create smooth, repeating waves that oscillate between -1 and 1. These functions help us model real-world phenomena like sound waves and planetary orbits.

Understanding how to graph sine and cosine is crucial. Key points, intercepts, and periodicity are essential concepts. Knowing the domain, range, and how transformations affect these graphs will give you a solid foundation for more advanced trigonometric applications.

Graphing Sine and Cosine Functions

Graphing sine and cosine functions

• Key points for sine function plot smooth curve through (0, 0), ($\frac{\pi}{2}$, 1), ($\pi$, 0), ($\frac{3\pi}{2}$, -1), ($2\pi$, 0)
• Key points for cosine function plot smooth curve through (0, 1), ($\frac{\pi}{2}$, 0), ($\pi$, -1), ($\frac{3\pi}{2}$, 0), ($2\pi$, 1)
• Intervals one complete cycle $[0, 2\pi]$ or $[-\pi, \pi]$, quarter-cycle intervals $[0, \frac{\pi}{2}]$, $[\frac{\pi}{2}, \pi]$, $[\pi, \frac{3\pi}{2}]$, $[\frac{3\pi}{2}, 2\pi]$
• Amplitude distance from midline to max or min point 1 for standard sine and cosine functions
• Period length of one complete cycle $2\pi$ for standard sine and cosine functions
• Sine and cosine graphs related by horizontal shift of $\frac{\pi}{2}$

Domain and range of trigonometric functions

• Domain all real numbers (-∞, ∞) in interval notation for both sine and cosine
• Range [-1, 1] for both sine and cosine functions limited by amplitude
• Continuous functions no breaks or gaps in graph smooth curve throughout domain
• Sine function oscillates between -1 and 1, passing through 0
• Cosine function oscillates between -1 and 1, starting at 1

Intercepts of sine and cosine

• Sine x-intercepts occur at multiples of $\pi$ (0, $\pi$, $2\pi$, ...)
• Cosine x-intercepts occur at odd multiples of $\frac{\pi}{2}$ ($\frac{\pi}{2}$, $\frac{3\pi}{2}$, ...)
• Sine y-intercept (0, 0) function passes through origin
• Cosine y-intercept (0, 1) function starts at maximum value
• General formulas for x-intercepts:
  1. Sine: $x = n\pi$, n is any integer
  2. Cosine: $x = \frac{\pi}{2} + n\pi$, n is any integer

Periodicity in trigonometric functions

• Periodicity function repeats values at regular intervals (sound waves)
• Period length of one complete cycle $2\pi$ for standard sine and cosine
• Frequency number of cycles completed in given interval inverse of period
• Transformations affecting periodicity:
  • Horizontal stretch/compression changes period
  • Vertical stretch/compression does not affect period
• Applications model cyclical phenomena (planetary orbits, tides, alternating current)