Chap 1 trigonometry 2 part 1

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.)


2

1.1 Graphs of Trigonometric Functions1.1.1 The Sine Curve


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


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.


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.


6

Example 1Example 1

Solution:

Step 1: Identify a = 5, b = 4, c = 0Therefore,▫ amplitude,

▫ Period,

▫ Phase shift,

55 a

24

22

b


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


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


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.


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 3: Construct a table and determine the values of

x & y.


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 3: Construct a table and determine the values of x &

y.


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| =



y.


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| =



y.


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



y.


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

Date post:	27-May-2015
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Chap 1 trigonometry 2 part 1

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