Nat 5
Trigonometric
Functions and Graphs
Graphs of the form y = a sin xo
Graphs of the form y = a sin bxo
Phase angle
Solving Trig Equations
Special trig relationships
Trigonometric
Functions and Graphs
Learning Intention
1. To identify key features of graphs of trigonometric
functions including:
y = a sin xo
y = a cos xo
y = a tan xo
Key Features
Sine Function Graph
Zeros at 0, 180 and 360
o
o
Maximum value at x = 90o
Minimum value at x = 270o
Key Features
Domain is 0 to 360o
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
Sine Function Graph
y = sinxo
y = 2sinxo
y = 3sinxo
3
y = 0.5sinxo
y = -sinxo
2
1
0
-1
-2
-3
90o
180o
270o
360o
Sine Function Graph
y = 5sinxo
y = 4sinxo
y = sinxo
6
y = -6sinxo
4
2
0
-2
-4
-6
90o
180o
270o
360o
Key Features
Cosine Function Graph
Zeros at 90 and 270
o
o
Maximum value at x = 0o and 360o
Minimum value at x = 180o
Key Features
Domain is 0 to 360o
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
y = cosxo
Cosine Function Graphy = 2cosx
o
y = 3cosxo
3
y = 0.5cosxo
2
y = -cosxo
1
0
-1
-2
-3
90o
180o
270o
360o
y = cosxo
Cosine Function Graph
y = 4cosx
o
y = 6cosxo
6
y = 0.5cosxo
y = -1.5cosxo
4
2
0
-2
-4
-6
90o
180o
270o
360o
Tangent Function
Graph
Zeros at 0 and 180
Key Features
o
Key Features
Undefined at 90o and 270o
Key Features
Domain is 0 to 180o
(repeats itself every 180o)
Tangent Function Graph
created by Mr. Lafferty
Cosine Function Graph
y = a sin (x)
y = a cos (x)
y = a tan (x)
For a > 1 stretches graph in the y-axis direction.
For a < 1 compresses graph in the y - axis direction.
For a < 0 graph reflects in the x – axis.
Trigonometric
Functions and Graphs
Learning Intention
1. To identify key features of graphs of trigonometric
functions including:
y = sin bxo
y = cos bxo
y = tan bxo
Period of a Function
When a pattern repeats itself over and over,
it is said to be periodic.
Sine function has a period of 360o
Cosine function has a period of 360o
Consider
and
y = sin bx
y = cos bx
Sine Function Graphy = sinx
o
y = sin2xo
y = sin4xo
3
y = sin0.5xo
2
1
0
-1
-2
-3
90o
180o
270o
360o
Cosine Function Graph
y = cosxo
y = cos2xo
3
y = cos3xo
2
1
0
-1
-2
-3
90o
180o
270o
360o
Period of a Function
When a pattern repeats itself over and over,
it is said to be periodic.
Tangent function has a period of 180o
Consider
y = tan bx
Tangent Function Graph
y = tanxo
Tangent Function Graph
y = tan2xo
Tangent Function Graph
y = tan3xo
Sine, Cosine & Tangent
Functions
y = a sin (bx)
y = a cos (bx)
y = a tan (bx)
b is how many times
graph repeats
itself in 360o
b is how many times
it repeats
itself in 180o
Trigonometric
Functions and Graphs
Learning Intentions
1. To identify key features of graphs of trigonometric
functions including:
y = asin bxo
y = acos bxo
y = atan bxo
2. To sketch graphs of trigonometric functions of this form.
Trigonometric Graphs
Write down an
equation for the
graph shown.
y = 0.5sin2xo
y = 2sin4xo
3
y = 3sin0.5xo
2
1
0
-1
-2
-3
90o
180o
270o
360o
Write down an
equation for the
graph shown.
y = 1.5cos2xo
Trigonometric Graphs y = -2cos2x
y = 0.5cos4xo
3
2
1
0
-1
-2
-3
o
90o
180o
270o
360o
Trigonometric
Functions and Graphs
Learning Intentions
1. To identify the phase angle in graphs of trigonometric
functions of the form:
y = a sin (x-b)o
2. To sketch graphs of trigonometric functions of the
form:
y = a sin (x-b)o
The Sine Function Graph
y = sin(x - 45)o
1
0
-1
45o
45o
90o
180o
270o
360o
The Sine Function Graph
y = sin(x + 60)o
1
60o
-60o
0
-1
90o
180o
270o
360o
By how much do we have
to move the ‘new’ cosine
curve so it fits on the
Int 2
original cosxo curve?
The Cosine Function Graph
y = cos(x - 70)o
1
0
-1
70o
o
90o 160 180o
270o
360o
By how much do we have
to move the ‘new’ cosine
curve so it fits on the
Int 2
original cosxo curve?
The Cosine Function Graph
y = cos(x + 56)o
1
0
-1
56o
34o
90o
180o
270o
360o
Phase Angle
y = sin (x - b)
y = cos (x - b)
b moves graph
along x – axis.
Naming a Function
y = a cos (x – b)
a =3 b = -30
y = 3 cos (x - 30)
By how much do we have
o curve
o and
o
to move
the
cosx
sinx
cosx
Similarly,
o = sinxo
cos(x+90)
o
o
so sin(x-90)
it fits
onto the
are
90 exactly
outo of
phase.
= cosx
sinxo curve?
Phase Angle & Graphs
1
0
-1
180o
360o
540o
720o