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CHAPTER 1: TRIGONOMETRY 2
Why study these trigonometric graphs?
Chapter 1 : Trigonometry 2
1
• The trigonometric graph are probably the most commonly use in all areas of science & engineering.
• They are used in modelling many different natural and mechanical phenomena (population, waves, engines, electronics, UV intensity, growth of plants & animal, etc.)
Chapter 1 : Trigonometry 2
2
1.1 Graphs of Trigonometric Functions1.1.1 The Sine Curve
Chapter 1 : Trigonometry 2
3
How does the sine curve look like?Let us consider the graph of the function f(x) = sin x.
180 Radians to Degree
180 Degree to Radians
Its easier to
calculate the values
in Deg mode
Chapter 1 : Trigonometry 2
4
1.1.2 The Cosine CurveHow does the cosine curve look like?Let us consider the graph of the function f(x) = cos x.
3 important term used in sketching a trigonometric graph : 1.1. PeriodPeriod : A function f is called PERIODIC if
there exists a ‘+’ real number p such that : f (x + p ) = f(x)
▫ The period of y = sin bx and y = cos bx where b > 0 is 2π/b
2.2. AmplitudeAmplitude: The maximum functional value of the graph. It is the coefficient of the trigo. functions.
▫ the amplitude of y = a sin x or y = a cos x, is |a| = a.
Chapter 1 : Trigonometry 2
5
3 important term used in sketching a trigonometric graph : 3.3. Phase shiftPhase shift: The shifting to the right or
to the left of a trigonometric curve is called the phase shift.
▫ For y = sin (x – c) or y= cos (x – c), the phase shift is |c|. For (x – c), the graph will shift to the RIGHT. For (x + c), the graph will shift to the LEFT.
Chapter 1 : Trigonometry 2
6
Example 1Example 1
Solution:
Step 1: Identify a = 5, b = 4, c = 0Therefore,▫ amplitude,
▫ Period,
▫ Phase shift,
55 a
24
22
b
Chapter 1 : Trigonometry 2
7
shiftphasenoc 0
Determine the period, amplitude and phase shift of y = 5 sin 4x.
y = 5 sin 4x.a
b
Example 2Example 2
Solution:
Step 1: Identify a = -2 , b = 1, c = 3Therefore,▫ amplitude,
▫ Period,
▫ Phase shift,
22 a
2
1
22
b
Chapter 1 : Trigonometry 2
8
.33 unitsbyrightthetoShiftc
Determine the period, amplitude and phase shift of y = -2 cos (x – 3).
y = -2 cos (x – 3).a
b
c
Example 3Example 3
Solution:
Therefore,▫ amplitude,
▫ Period,
▫ Phase shift,
44 a
3
22
b
Chapter 1 : Trigonometry 2
9
.22 unitsbyleftthetoShiftc
Determine the period, amplitude and phase shift of y = -4 sin 3(x + 2).
y = -4 sin 3(x + 2).a
b
c
Solution:Step 1: From y = cos 3x; a = 1, b = 3therefore, period = 2π/3 = 120 ° & |a| = 1
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of
x & y.
Chapter 1 : Trigonometry 2
10
Determine the period of y = cos 3x and sketch the graph of one period beginning at x = 0.
Example 4Example 4
304
0120I
x 0° 30° 60° 90° 120°
y = cos 3x
1 0 -1 0 1
Solution:Step 1: From y = -3 sin 0.5x ; b = 0.5therefore, period = 2π/0.5 = 2π= 720° & |a| = 3
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of
x & y.
Chapter 1 : Trigonometry 2
11
Determine the period and amplitude of y = -3 sin 0.5x and sketch the graph of one period beginning at x = 0.
Example 5Example 5
1804
0720I
x 0° 180° 360°
540°
720°
sin 0.5x
0 1 0 -1 0
-3 sin 0.5x
0 -3 0 3 0
Solution:Step 1: From y = 3 sin (x- π); a= 3, b = 1therefore, period = 2π/1= 2π = 360° & |c| = π, shift to the right
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of x &
y.
Chapter 1 : Trigonometry 2
12
Determine the period, amplitude and the phase shift of y = 3 sin (x- π) and sketch the graph for
Example 6Example 6
904
0360I
x 0° 90° 180°
270°
360°
(x-π)(x-
180°)
-180°
-90° 0° 90° 180°
3sin (x-π)
0 -3 0 3 0
20 x
Solution:Step 1: |a|= , b = period = , & |c| =
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of x &
y.
Chapter 1 : Trigonometry 2
13
Determine the period, amplitude and the phase shift of y = 2 sin (x + π/2) and sketch the graph for
Example 7Example 7
x 0° 90° 180°
270°
360°
20 x
Try Ex 5 pg 9
Solution:Step 1: |a|= , b = period = , & |c| =
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of x &
y.
Chapter 1 : Trigonometry 2
14
Determine the period, amplitude and the phase shift of y = 2.5 cos(3x –π) and sketch the graph for
Example 8Example 8
x
20 x
y = 2.5 cos 3(x – π/3).Factorize
b
Solution:Step 1: From y = 2+3 sin (x- π); a= 3, b = 1, d = 2therefore, period = 2π/1= 2π = 360° & |c| = π, shift to the right
Step 2: Determine the subinterval,
Step 3: Construct a table and determine the values of x &
y.
Chapter 1 : Trigonometry 2
15
Determine the period, amplitude and the phase shift of y = 2+3 sin (x- π) and sketch the graph for
Example 9Example 9
904
0360I
x 0° 90° 180°
270°
360°
(x-π) -180°
-90° 0° 90° 180°
3sin (x-π)
0 -3 0 3 0
2+3sin (x-
π)
2 -1 2 5 2
20 x
Try Tut 1 pg 203 -
205
SummarySummary• For the function y = a sin b (x – c) or y = a cos b (x – c) where
b>0:
# The period is for all values of x.
# The amplitude is |a| for all values of x.
# The phase shift is |c|.For (x – c), the graph will shift to the right.For (x + c), the graph will shift to the left.
# The displacement is |d|.For +d , the graph will displace upside.For -d, the graph will shift displace downside.
b
2
16