The Graphs of Sine, Cosine and Tangent Functions

Back

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: