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Chap 1 trigonometry 2 part 1
 

Chap 1 trigonometry 2 part 1

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    Chap 1 trigonometry 2 part 1 Chap 1 trigonometry 2 part 1 Presentation 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.