Geometry and Trigonometry

The circular functions sin x, cos x, and tan x represent the fundamental relationships found in a right-angled triangle, extended to all real numbers using the unit circle. They are periodic functions, which means they repeat their values in regular intervals. The amplitude of these functions indicates the maximum value they reach, and this periodic nature allows us to model many real-life phenomena such as sound waves or seasonal patterns. When we graph these functions, they create a wave-like pattern, each with distinctive shapes corresponding to their ratios in the unit circle. Composite functions of the form \( f(x) = a \sin(b(x + c)) + d \) allow for adjustments in amplitude, frequency, phase shift, and vertical shift, enabling a more precise modelling of real-world situations.

Composite trigonometric functions can be transformed to fit specific criteria, useful in both theoretical and practical applications. The 'a' value modifies the amplitude, 'b' affects the period of the function, 'c' adjusts the horizontal shift, and 'd' translates the function vertically. These transformations are essential in applications ranging from physics to engineering, where waves and oscillations describe many different phenomena.

Example: To model the daily hours of daylight over a year at a particular latitude, one might use the function:

$$ f(x) = A \sin\left(\frac{2\pi}{365}(x - \phi)\right) + D $$

where:

\( A \) is the amplitude, representing the maximum deviation from the average daylight hours,
\( B = \frac{2\pi}{365} \) represents the period of the function, reflecting the number of days in a year,
\( \phi \) is the phase shift, corresponding to the day with the least or most daylight,
\( D \) is the average daylight hours over the year.

This video on Trig Graphs and Circular Functions is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.