Trigonometry Introduction and Periodic Functions
Introduction to Trigonometry
Importance of Trigonometry
- Trigonometry is essential because it models phenomena that occur periodically, meaning they repeat over and over.
Periodic Functions
- Definition: A periodic function is one that repeats its values at regular intervals.
- Example: The amount of daylight throughout the year.
- In the Northern Hemisphere, daylight hours are shorter in winter and longer in summer.
- Plotting daylight hours daily results in a repeating pattern.
- Period: The length of one complete cycle of the function.
- It's the distance from one peak to the next peak (or from valley to valley) on the graph of the function.
- The period represents how long it takes for the function to repeat itself.
Functions Previously Studied
- The course has covered various types of functions:
- Linear functions
- Quadratic functions
- Polynomial functions
- Radical functions
- Rational functions (with asymptotes)
- Exponential functions
- Logarithmic functions
- None of these functions repeat themselves like trigonometric functions do.
Calculating the Period
- The period can be measured from any point on the function to the corresponding point in the next cycle.
- Example: A function with irregular peaks and valleys can still be periodic if a section of it repeats.
- If a point (2, 5) corresponds to a point (9, 5) on the next cycle, the period is the difference in the x-coordinates.
- Period =
Key Concepts
- Periodic Function: A function that repeats itself over regular intervals.
- Period: The distance (or time) it takes for the function to complete one full cycle and begin repeating.
- Finding the horizontal distance between any two corresponding points on the graph. This distance will remain consistent throughout the function.
This is an introduction to periodic functions and how the periods are calculated.