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# Chap 1 trigonometry 2 part 1

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### Chap 1 trigonometry 2 part 1

1. 1. CHAPTER 1: TRIGONOMETRY 2 Why study these trigonometric graphs? Chapter 1 : Trigonometry 2
2. 2. <ul><li>The trigonometric graph are probably the most commonly use in all areas of science & engineering. </li></ul><ul><li>They are used in modelling many different natural and mechanical phenomena (population, waves, engines, electronics, UV intensity, growth of plants & animal, etc.) </li></ul>Chapter 1 : Trigonometry 2
3. 3. 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
4. 4. 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.
5. 5. 3 important term used in sketching a trigonometric graph : <ul><li>Period : A function f is called PERIODIC if there exists a ‘ + ’ real number p such that : </li></ul><ul><li>f ( x + p ) = f ( x ) </li></ul><ul><ul><li>The period of y = sin bx and y = cos bx where b > 0 is 2 π /b </li></ul></ul><ul><li>Amplitude : The maximum functional value of the graph. It is the coefficient of the trigo. functions. </li></ul><ul><ul><li>the amplitude of y = a sin x or y = a cos x , is | a | = a . </li></ul></ul>Chapter 1 : Trigonometry 2
6. 6. 3 important term used in sketching a trigonometric graph : <ul><li>Phase shift : The shifting to the right or to the left of a trigonometric curve is called the phase shift . </li></ul><ul><ul><li>For y = sin (x – c) or y= cos (x – c), the phase shift is | c |. </li></ul></ul><ul><ul><ul><li>For ( x – c ), the graph will shift to the RIGHT . </li></ul></ul></ul><ul><ul><ul><li>For ( x + c ), the graph will shift to the LEFT. </li></ul></ul></ul>Chapter 1 : Trigonometry 2
7. 7. Example 1 <ul><li>Solution: </li></ul><ul><li>Step 1: Identify a = 5, b = 4, c = 0 </li></ul><ul><li>Therefore, </li></ul><ul><ul><li>amplitude, </li></ul></ul><ul><ul><li>Period, </li></ul></ul><ul><ul><li>Phase shift, </li></ul></ul>Chapter 1 : Trigonometry 2 Determine the period, amplitude and phase shift of y = 5 sin 4 x . y = 5 sin 4 x . a b
8. 8. Example 2 <ul><li>Solution: </li></ul><ul><li>Step 1: Identify a = -2 , b = 1, c = 3 </li></ul><ul><li>Therefore, </li></ul><ul><ul><li>amplitude, </li></ul></ul><ul><ul><li>Period, </li></ul></ul><ul><ul><li>Phase shift, </li></ul></ul>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
9. 9. Example 3 <ul><li>Solution: </li></ul><ul><li>Therefore, </li></ul><ul><ul><li>amplitude, </li></ul></ul><ul><ul><li>Period, </li></ul></ul><ul><ul><li>Phase shift, </li></ul></ul>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
10. 10. <ul><li>Solution: </li></ul><ul><li>Step 1: From y = cos 3 x; a = 1, b = 3 </li></ul><ul><li>therefore, period = 2 π /3 = 120 ° & |a| = 1 </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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
11. 11. <ul><li>Solution: </li></ul><ul><li>Step 1: From y = -3 sin 0.5 x ; b = 0.5 </li></ul><ul><li>therefore, period = 2 π /0.5 = 2 π = 720 ° & |a| = 3 </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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
12. 12. <ul><li>Solution: </li></ul><ul><li>Step 1: From y = 3 sin (x- π ) ; a= 3, b = 1 </li></ul><ul><li>therefore, period = 2 π /1= 2 π = 360 ° & |c| = π , shift to the right </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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
13. 13. <ul><li>Solution: </li></ul><ul><li>Step 1: |a|= , b = </li></ul><ul><li>period = , & |c| = </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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°
14. 14. <ul><li>Solution: </li></ul><ul><li>Step 1: |a|= , b = </li></ul><ul><li>period = , & |c| = </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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
15. 15. <ul><li>Solution: </li></ul><ul><li>Step 1: From y = 2+3 sin (x- π ) ; a= 3, b = 1, d = 2 </li></ul><ul><li>therefore, period = 2 π /1= 2 π = 360 ° & |c| = π , shift to the right </li></ul><ul><li>Step 2: Determine the subinterval, </li></ul><ul><li>Step 3: Construct a table and determine the values of x & y . </li></ul>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
16. 16. Summary <ul><li>For the function y = a sin b (x – c) or </li></ul><ul><li>y = a cos b (x – c) where b>0: </li></ul><ul><li># The period is for all values of x. </li></ul><ul><li># The amplitude is |a| for all values of x. </li></ul><ul><li># The phase shift is |c|. </li></ul><ul><li>For (x – c), the graph will shift to the right. </li></ul><ul><li>For (x + c), the graph will shift to the left. </li></ul><ul><li># The displacement is |d|. </li></ul><ul><li>For +d , the graph will displace upside. </li></ul><ul><li>For -d, the graph will shift displace downside. </li></ul>