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: