360

1
Higher Maths 1 2 3
Trigonometric Functions
1
UNIT
OUTCOME
SLIDE
PART
NOTE
Trigonometric Graphs
y
amplitude
Graphs of trigonometric equations are wave shaped
with a repeating pattern.
720
360
x
y sin x
Amplitude
period
Half of the vertical height.
Graphs of the tangent function
Period
the amplitude cannot be measured.
y
The horizontal width of one wave section.
x
y tan x
2
Higher Maths 1 2 3
Trigonometric Functions
2
UNIT
OUTCOME
SLIDE
PART
NOTE
Example
Amplitude and Period
amplitude
For graphs of the form
a 3
y
y a sin bx c
y 3 cos 5 x 2
y a cos bx c
and
c 2
x
amplitude a
360
period
period
360
b
72
5
For graphs of the form
y
180
period
b
x
y a tan bx c
(amplitude is undefined)
3
Higher Maths 1 2 3
Trigonometric Functions
3
UNIT
OUTCOME
SLIDE
PART
r
NOTE
r
Radians
r
r
Angles are often measured in radians instead of
degrees.
60
r
r
one radian
A radian is not 60
A radian is the angle for which the length of the
arc is the same as the radius.
r
r
The radius fits into the circumference
times.
C p D
2 p
r
C 2 p r
r
r
r
2 p
Radians are normally written as fractions of .

6.28
p
Inaccurate
4
Higher Maths 1 2 3
Trigonometric Functions
4
UNIT
OUTCOME
SLIDE
PART
NOTE
Exact Values of Trigonometric Functions





0
30
45
60
90
x
p
p
p
p
0
2
3
4
6
sin x
0
1
cos x
1
0
not defined
tan x
1
0
5
Higher Maths 1 2 3
Trigonometric Functions
5
UNIT
OUTCOME
SLIDE
PART
NOTE
Quadrants
It is useful to think of angles in terms of
quadrants.
p
Examples
90
2
1st
2nd
1st
2nd
37
p
180
0
0
4th
3rd
4th
3rd
270
37 is in the 1st quadrant
6
Higher Maths 1 2 3
Trigonometric Functions
6
UNIT
OUTCOME
SLIDE
PART
NOTE
The Quadrant Diagram
90
1st
2nd
The nature of trigonometric functions can be
shown using a simple diagram.
sin cos tan
sin cos tan
y
1st
2nd
4th
3rd
180
0
sin cos tan
sin cos tan
x
3rd
4th
270
360
270
180
90


S
A
The Quadrant Diagram


T
C
all positive
sin positive
tan positive
cos positive
7
Higher Maths 1 2 3
Trigonometric Functions
7
UNIT
OUTCOME
SLIDE
PART
NOTE
Quadrants and Exact Values
Any angle can be written as an acute angle
starting from either 0 or 180.
tan negative
120
225
60
45
cos negative


sin 120
cos 225
-
- cos 45
sin 60


8
Higher Maths 1 2 3
Trigonometric Functions
8
UNIT
OUTCOME
SLIDE
PART
NOTE
Solving Trigonometric Equations Graphically
It is possible to solve trigonometric equations
by sketching a graph.
Example
2p
2 cos x 3 0
0 x
Solve
for
v
2 cos x
3
v
cos x
y
Sketching y cos x gives
p
p
x
2p
x
p
x
11p
p
2p
or
6
6
6
6
11p

y cos x
6
9
Higher Maths 1 2 3
Trigonometric Functions
9
UNIT
OUTCOME
SLIDE
PART
NOTE
Solving Trigonometric Equations using Quadrants
Trigonometric equations can also be solved
algebraically using quadrants.
The X-Wing
Example
Diagram
2 sin x 1 0
Solve
v
0 x
for
360
solutions are in the 3rd and 4th quadrants
-
sin x
45
45
P
P
sin negative
acute angle
x 180 45
x 360 45
or
225
315
10
Higher Maths 1 2 3
Trigonometric Functions
10
UNIT
OUTCOME
SLIDE
PART
NOTE
(continued)
Solving Trigonometric Equations using Quadrants
Example 2
p
2
tan 4 x 3 0
Solve
P
v
p
0 x
solutions are in the 2nd and 4th quadrants
for
p
0
2
tan 4 x 3
-
v
P
3p
tan negative
2
p
p
p
2p
4 x
4 x
or
acute angle
3
3
p
2p
5p
(
)

tan
3
v


3
3
3
p
5p
x
x
6
12
11
Higher Maths 1 2 3
Trigonometric Functions
11
UNIT
OUTCOME
SLIDE
PART
NOTE
Dont forget to include angles more than 360
Problems involving Compound Angles
Example
6 sin ( 2 x 10 ) 3
Solve
P
P
solutions are in the 1st and 2nd quadrants
0 x
for
360
30
30
1
sin ( 2 x 10 )
2
360150
Consider the range
36030
2 x 10 30 or 150 or 390 or 510
0 x
360
2 x 20 or 140 or 380 or 500
0 2 x
720
x 10 or 70 or 190 or 250
10 2 x 10
730
12
Higher Maths 1 2 3
Trigonometric Functions
12
UNIT
OUTCOME
SLIDE
PART
NOTE
Solving Quadratic Trigonometric Equations
( sin x ) 2
is often
Example
sin 2 x
written
7 sin 2 x 3 sin x 4 0
Solve
0 x
for
360
( 7 sin x 4 ) ( sin x 1 ) 0
sin x 1 0
7 sin x 4 0
or
-
sin x 1
4
S
A
sin x
7
T
C
x 90
P
P
acute angle 34.8
x 180 34.8
x 360 34.8
or
214.8
325.2