The Graphs of Sine, Cosine and Tangent Functions

 6.2 The Graphs of Sine, Cosine and Tangent Functions

The characteristics of sine graph, $$y = \sin x$$:

• has a maximum value of 1 when $$x = 90^\circ$$ and a minimum value of –1 when $$x = 270^\circ$$
• intercepts $$x$$-axis at $$x = 0^\circ,180^\circ,360^\circ$$ ($$x$$-intercept)
• intercepts -axis at $$y=0$$ ( $$y$$-intercept)

The characteristics of cosine graph, $$y = \cos x$$:

• has a maximum value of 1 when $$x = 0^\circ$$ and $$x = 360^\circ$$ and a minimum value of –1 when $$x = 180^\circ$$
• intercepts $$x$$-axis at ($$x$$-intercept)
• intercepts $$y$$-axis at y = 1 ($$y$$-intercept)

The characteristics of tangent graph, $$y = \tan x$$:

• maximum value is $$\infty$$ and minimum value is $$-\infty$$
• intercepts $$x$$-axis at $$x = 0^\circ,180^\circ,360^\circ$$ ($$x$$-intercept)
• intercepts $$y$$-axis at $$y=0$$ ( $$y$$-intercept)
• the values of  $$\tan90^\circ$$ and $$\tan 270^\circ$$ are undefined

 The effects of changes in constants a, b and c on the graphs of trigonometric functions y = a sin bx + c, y = a cos bx + c and y = a tan bx + c

 Example 5 The diagram below shows a graph of the function $$y = \sin x \text{ for } 0^\circ \leq x \leq 360^\circ$$. Sketch each of the following trigonometric functions on the same axes. (a) $$y = 2 \sin x$$         (b) $$y=\sin 2x$$ (c) $$y = \sin x + 1$$        (d) $$y = 2 \sin 2x$$ Solution: