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Chap 1 trigonometry 2 part 1

on Jul 28, 2011

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Chap 1 trigonometry 2 part 1Presentation Transcript

• CHAPTER 1: TRIGONOMETRY 2 Why study these trigonometric graphs? Chapter 1 : Trigonometry 2
• 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
• 1.1 Graphs of Trigonometric Functions 1.1.1 The Sine Curve Chapter 1 : Trigonometry 2 How does the sine curve look like? Let us consider the graph of the function f ( x ) = sin x. Its easier to calculate the values in Deg mode
• Chapter 1 : Trigonometry 2 1.1.2 The Cosine Curve How 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 :
• Period : 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 2 π /b
• Amplitude : 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
• 3 important term used in sketching a trigonometric graph :
• Phase 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
• Example 1
• Solution:
• Step 1: Identify a = 5, b = 4, c = 0
• Therefore,
• amplitude,
• Period,
• Phase shift,
Chapter 1 : Trigonometry 2 Determine the period, amplitude and phase shift of y = 5 sin 4 x . y = 5 sin 4 x . a b
• Example 2
• Solution:
• Step 1: Identify a = -2 , b = 1, c = 3
• Therefore,
• amplitude,
• Period,
• Phase shift,
Chapter 1 : Trigonometry 2 Determine the period, amplitude and phase shift of y = -2 cos (x – 3). y = -2 cos (x – 3) . a b c
• Example 3
• Solution:
• Therefore,
• amplitude,
• Period,
• Phase shift,
Chapter 1 : Trigonometry 2 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 3 x; a = 1, b = 3
• therefore, period = 2 π /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 Determine the period of y = cos 3x and sketch the graph of one period beginning at x = 0 . Example 4 x 0° 30° 60° 90° 120° y = cos 3x 1 0 -1 0 1
• Solution:
• Step 1: From y = -3 sin 0.5 x ; b = 0.5
• therefore, period = 2 π /0.5 = 2 π = 720 ° & |a| = 3
• Step 2: Determine the subinterval,
• Step 3: Construct a table and determine the values of x & y .
Chapter 1 : Trigonometry 2 Determine the period and amplitude of y = -3 sin 0.5x and sketch the graph of one period beginning at x = 0 . Example 5 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 = 1
• therefore, period = 2 π /1= 2 π = 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 Determine the period, amplitude and the phase shift of y = 3 sin (x- π ) and sketch the graph for Example 6 x 0° 90° 180° 270° 360° (x- π ) (x-180 °) -180° -90° 0° 90° 180° 3sin (x- π ) 0 -3 0 3 0
• 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 Determine the period, amplitude and the phase shift of y = 2 sin (x + π /2) and sketch the graph for Example 7 Try Ex 5 pg 9 x 0° 90° 180° 270° 360°
• 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 Determine the period, amplitude and the phase shift of y = 2.5 cos(3x – π ) and sketch the graph for Example 8 y = 2.5 cos 3(x – π / 3) . Factorize b x
• Solution:
• Step 1: From y = 2+3 sin (x- π ) ; a= 3, b = 1, d = 2
• therefore, period = 2 π /1= 2 π = 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 Determine the period, amplitude and the phase shift of y = 2+3 sin (x- π ) and sketch the graph for Example 9 Try Tut 1 pg 203 - 205 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
• Summary
• 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.