6.3 graph of sine, cosine and tangent functions

6.3 graph of
sine, cosine,
and tangent

functions

HR MISS
SYAFAWATI

Radius Degree
2π 360
3π 270
2
π 180
π 90
2
π 45
4

Things you should know!!!

Graph y=sin x for -2π<x>2π

a) Amplitude=1

i) The maximum value is 1 while the minimum value is -1, so the amplitude of the graph is
1 unit.

b) period = 360° or 2π

c) x-intercepts: -2π, -π,0,π,2π

d) y-intercepts:0

e) The graph repeat itself every 360° or 2π rad

Cosine graph

Things you should know!!!

Graph y= cos x as for -2π ≤ x ≤ 2π
a) Amplitude= 1
b) The maximum value is 1 while the minimum value is -1
c) Period of graph is 360◦ / 2π
d) The line graph for x-intercepts fall on -π/2, π/2 , 3π/2
e) The line graph for y-intercepts fall at 1

WHAT?? IS THE DIFFERENECE BETWEEN :
SINE GRAPH AND COSINE GRAPH
SINE GRAPH:

• Line graph must pass through the origin , 0
COSINE GRAPH:

• Line graph can pass through any point except for the
origin,0

y = tan x graph

• Graph y = tan x is for -2 ≤ x ≤ 2π EXTRA!
a) No amplitude !
i) There is no maximum value of y
ii) There is no minimum value of y The general tangent
b) Period = 180° or π function is y = a tan
c) X-asymptotes : -3/2π, -1/2π, 1/2π, 3/2π bx + c, a>0, b>0 ;
d) X-intercepts : -2π, -π, 0, π, 2π
e) Y-intercepts : 0 the principle axis is
y+c.

the period of this
function is π/b.

the amplitude of this
function is undefined

• The graph y = tan x is not sinusoidal. The proper of y = tan x are as follows:
a) This graph has no maximum value.
b) The graph repeats itself every 180° or π rad interval, so the period of a tangent
graph is 180° or π rad.
c) The function y = tan x is not defined at x = 90° and x = 270°. The curve
approaches the line x = 90° and x = 270° but does touch the line. This line called
asymptote.





EXAMPLE 12

State the cosine function represented by the graph in the diagram below.

SOLUTION:
Note that the amplitude is 4.
So, a = 4.
Two cycles in the range of 0 ≤ x ≤ 2x.
The period is x, that is, 2x/b =. X, so b = 2.
Hence, the graph represents y = 4 cos 2x.

NOTE:
Besides identifying the trigonometric function of a given graph, the values of
Constants a, b and c also help in sketching graphs when the trigonometric
Functions are given.







Example 14 (a) Reflection o
be followed
get graph y2

State the transformation on the function graph y = tan x to
obtain each of the following graphs.
(b) The reflecti

(a) y = |–tan x| graph y1 = |

(b) y = –|tan x|

(c) Then, sketch both graphs for 0 ≤ x ≤ 2π.

solution Y = tan x

period = π rad

Y1 = |tan x|

of graph y = tan x on x-axis results in getting graph y1 = -tan x to
d by a reflection of negative part of graph y1 = -tan x on x-axis to
2 = |-tan x|.
ion of negative part of graph y = tan x on x-axis results in getting
|tan x| to be followed by reflection on x-axis to get y2 = -|tan x|.

Y1= -tan x

Y2 = |-tan x|

Y2 = -|tan x|

Example 15

(15)

On the same axes, draw the graphs y = sin 2x and y =x/2x - for 0 ≤ ≤ .
Then, state the
solutions to the trigonometric equation 2 sin 2 =0

The graphs y=sin 2 and y=
2

The points of intersection of the two graphs are the solutions to sin 2 = or

2

2 sin 2 − = 0

From the graph, it is found that the solutions to the equation 2 sin 2 − =
0 are 0 and 0.46