Transcript y x

Higher Maths
1
Trigonometric Graphs
2
3
Trigonometric Functions
1
amplitude
Graphs of trigonometric
equations are wave shaped
with a repeating pattern.
Amplitude
Half of the vertical height.
360°
period
720°
y = sin x
Graphs of the tangent function:
Period
the amplitude
cannot be
measured.
The horizontal width of
one wave section.
y = tan x
Higher Maths
1
Amplitude and Period
2
and
amplitude =
period =
Trigonometric Functions
2
Example
amplitude
For graphs of the form
y = a sin bx + c
y = a cos bx + c
3
a=3
y = 3 cos 5 x + 2
c=2
a
360°
b
For graphs of
the form
y = a tan bx + c
period
360°
= 72°
5
period =
180°
b
(amplitude is undefined)
Higher Maths
1
2
3
Trigonometric Functions
r
Radians
one
radian
A radian is the angle for which
the length of the arc is the same
as the radius.
r
C =
C =
2
π
πD
2π r
≈ 6.28…
r
r
Angles are often measured
in radians instead of degrees.
r
r
r
r
r
r
Inaccurate
60°
r
A radian
is not 60°
The radius fits into the
circumference 2 times.
π
360° degrees = 2
r
3
π radians
Radians are normally written
as fractions of .
π
Higher Maths
1
2
3
Trigonometric Functions
Exact Values of Trigonometric Functions
x
sin x
cos x
tan x
0°
30°
45°
60°
90°
0
π
π
π
π
0
1
2
1
√2
√3
1
1
√3
1
√2
1
2
0
0
1
√3
1
√3
not
defined
6
2
4
3
2
2
4
Higher Maths
1
2
3
Trigonometric Functions
5
Quadrants
It is useful to think of angles in terms of quadrants.
Examples
π
90°
2nd
1st
37°
180°
3rd
2nd
0°
37° is in the 1st quadrant
π
1st
4
3
π
4th
270°
2
3rd
π
0
π
3
2
4th
4
is in the 3rd quadrant
3
Higher Maths
1
2
3
Trigonometric Functions
The Quadrant Diagram
The nature of trigonometric functions can
be shown using a simple diagram.
1st
2nd
90°
3rd
180°
4th
270°
sin +
cos –
tan –
180°
3rd
360°
+
S
all
sin
tan
cos
positive positive positive positive
90°
2nd
+
T
sin –
cos –
tan +
sin +
cos +
tan +
sin –
cos +
tan –
270°
6
1st
0°
4th
+
A
+
C
The Quadrant
Diagram
Higher Maths
1
2
3
Trigonometric Functions
7
Quadrants and Exact Values
Any angle can be written as an acute angle
starting from either 0° or 180°.
S+
+
S+
A
225°
120°
60°
+
A
T
+
C
sin 120° =
+
cos T
negative
+
C
cos 225° =
3
sin 60° = √
2
1
- cos 45° = √2
A
π
6
45°
+
+
tan S+
negative
- 76π
+
T
π
tan - 7
6
- tan π
6
+
C
=
=
-
1
√3
Higher Maths
1
2
3
Trigonometric Functions
8
Solving Trigonometric Equations Graphically
It is possible to solve trigonometric equations by sketching a graph.
Example
Solve 2 cos x – √ 3 = 0 for 0
x
2π
2 cos x = √ 3
cos x =
Sketching
x
π
= 6
y = cos x
or
x
√3
2
√3
gives:
2
π
= 2π – 6
11π
= 6
π
6
π
11π 2π
6
y = cos x
Higher Maths
1
2
3
Trigonometric Functions
Solving Trigonometric Equations using Quadrants
Trigonometric equations can also be solved
algebraically using quadrants.
The ‘X-Wing’
Example
Diagram
Solve √ 2 sin x + 1 = 0
for 0°
x
S+
360°
sin x =
- √12
45°
sin negative
acute angle:
sin-1
1
√2
( ) = 45°
A+
x
45°
T+
P
+
C
P
= 180° + 45°
= 225°
or
9
S+
A+
T+
C
+
solutions are
in the 3rd and 4th
quadrants
x
= 360° – 45°
= 315°
Higher Maths
1
2
3
Trigonometric Functions
10
Solving Trigonometric Equations using Quadrants (continued)
Example 2
π
Solve tan 4 x + √ 3 = 0
for 0
x
π
2
S+
P
A+
π
2
tan 4 x =
-√ 3
0
T+
π
3
2
tan negative
acute angle:
4x =
π
tan-1(√ 3 ) = 3
=
x
=
π – π3
2π
+
C
P
or
solutions are
in the 2nd and 4th
quadrants
π
4 x = 2π – 3
=
3
π
6
x
=
5π
3
5π
12
Higher Maths
1
2
3
Trigonometric Functions
Don’t forget to include
angles more than 360°
Problems involving Compound Angles
Example
Solve 6 sin ( 2 x + 10 ) = 3
for 0°
x
360°
sin ( 2 x + 10 ) =
1
2
Consider
the range:
0°
x
0°
2x
10°
A+P
T+
C
30°
solutions are
in the 1st and 2nd
quadrants
+
360°+30°
360°+150°
2 x + 10 = 30° or 150° or 390° or 510°
360°
2 x = 20° or 140° or 380° or 500°
720°
2 x + 10
S +P
30°
11
730°
x
= 10° or 70° or 190° or 250°
Higher Maths
1
2
3
Trigonometric Functions
Solving Quadratic Trigonometric Equations
Example
( sin x ) 2 is often
written sin 2 x
Solve 7 sin 2 x + 3 sin x – 4 = 0
for 0°
x
360°
( 7sin x + 4 ) ( sin x – 1 ) = 0
7sin x + 4 = 0
sin x =
-
or
4
7
acute angle ≈ 34.8°
x
≈ 180° + 34.8°
≈ 214.8°
S
A
T
C
sin x – 1 = 0
sin x = 1
x
P P
or
x
≈ 360° – 34.8°
≈ 325.2°
= 90°
12