Maths - Trigonometry

Periodic functions of the type y = a sin px

The graph of y = a sin(px) is similar to that of the sin graph of y = sin(x). Instead of varying or oscillating between +1 and -1, a sin(px) will vary between +a and -a. 'a' is called the amplitude and affects only the scale for y. The graph will repeat itself when the angle px increases by 360° or 2π radians. i.e. when x increases by 360°/p or 2π/p radians thus the period is 2π/p radians.

sine wave

Thus 5 sin(2x) has a period of 2π/2 = π radians and amplitude 5. Hence there are 2 waves between 0° and 360°.

Example

Sketch the curve for y = 3 sin(x+60°)

phase

y = 3 sin(x+60°) is said to have a phase lead of 60° since its values are those of y = 3 sin(x) with x increased by 60°. y = 3 sin(x-60°) is said to lag by 60°.

Solving Trig Equations

Here are some suggestions for solving trig identities:

  1. Change secant, cosecant and cotan into cos, sin and tan.
  2. rewrite in terms of sin and cos (use tan = sin/cos)
  3. If in second degree (ie cos2 or sin2) look for posible cos2A + sin2A = 1 relationship.
  4. If it contains cos2 or sin2 change sin2 into cos2-1 and solve resulting quadratic.
  5. If all this fails convert to an infinite series.
  6. If this fails plot as a graph.

Plotting trig functions

Example 1

y = 3 sin(2x) + sin x

trig example 1

Example 2

y = 3 sin(x/2) - cos x

trig example 2


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