Trigonometric Graphs

Understanding Trigonometric Graphs

  • A trigonometric graph is a visual representation of a trigonometric function such as sine, cosine, and tangent.
  • These graphs exhibit specific patterns and can model phenomena in physics and engineering, as well as represent cyclical occurrences like the seasons or wave patterns.

Recognising Trigonometric Functions

  • The sine function (sin) starts from the origin (0,0), peaks at (90,1), decreases to (180,0), bottoms out at (270,-1), and returns to (360,0). This pattern repeats every 360 degrees.
  • The cosine function (cos) starts at (0,1), decreases to (90,0), bottoms out at (180,-1), increases to (270,0) and returns to (360,1). Like the sine function, this pattern repeats every 360 degrees.
  • The tangent function (tan) also starts from the origin, but its graph is not confined between -1 and 1. It instead exhibits vertical asymptotes, or undefined areas, at 90 degrees, -90 degrees and every 180 degrees thereafter.

Amplitude and Period of Trigonometric Graphs

  • The amplitude of a trigonometric graph is the height of the graph from the centre line to its peak or trough. For standard sine and cosine graphs, this is 1.
  • The period is the length (in degrees or radians) of one complete wave. For all standard trigonometric graphs, this is 360 degrees or 2*Pi radians.

Transformations of Trigonometric Graphs

  • Just like any other graph, trigonometric graphs can be subject to translation, reflection and stretching.
  • Translations move the graph vertically (up or down) or horizontally (left or right). In sine and cosine graphs, horizontal translations change the phase of the cycle.
  • Reflections flip the graph over the horizontal or vertical axis, changing the sign of the function.
  • Stretching can affect the graph vertically (changing the amplitude) or horizontally (changing the period).

Sketching Trigonometric Graphs

  • To sketch a trigonometric graph, start by marking out key points according to the nature of the function and the transformations in play.
  • Consider amplitude, period, phase shift (horizontal translation), and vertical translation.
  • The more key points plotted, the more accurate your sketch of the curve between the points will be.

Mastering trigonometric graphs requires understanding the nature of the trigonometric functions and the effects of transformations. Practice with different functions and transformations will yield clearer understanding over time.