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Starter. Find all values of Θ between 0 0 and 360 0 for which 1. cosΘ = √3 / 2 Cos Θ is positive in the 1 st or 4 th quadrant Θ = 30 0 , 330 0 2. Sin Θ + ½ = 0 Sin Θ = -½ SinΘ is negative in the 3 rd and 4 th quadrant Θ = 210 0 , 330 0.

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  1. Starter Find all values of Θ between 00 and3600 for which 1. cosΘ = √3/2 Cos Θ is positive in the 1st or 4th quadrant Θ = 300 , 3300 2. Sin Θ + ½ = 0 Sin Θ = -½ SinΘ is negative in the 3rd and 4th quadrant Θ = 2100, 3300

  2. Note 4: Sine and Cosine Curve Draw an accurate sketch of the Sine and Cosine Curve: y = sin x y = cos x for -3600 ≤ Θ ≤3600 x-axis -plot every 15° y-axis from -1 to 1 – plot every 0.1

  3. Sine Curve y = sin x (90, 1) 1 360 180 – 1 Switch your calculator to either radiansor degrees (Remember 180° =  radians) 450

  4. Characteristics of the Sine and Cosine Curve • The period is 360° ( how long it takes for the graph to repeat • The amplitude is 1 ( the height from the middle • The maximum value is 1 and minimum value is -1 • The domain is: -360° < x < 360° • The range is: -1 < y < 1 • The cosine curve is just the sine curve shifted by 90°

  5. AMPLITUDE, A y = sin x (90, 1) 1 180 360 540 – 1 The amplitudeis equal to 1. This is the vertical distance from the equilibrium line (the x-axis in this case) to a peak or trough

  6. y = sin x (/2, 1) 1 0  2 3 – 1 PERIOD T = 360/B (or 2/B if in Raidans) The Period(aka wavelength) is the horizontal distance between points where the curve repeats itself. Period equals 2 (or 3600) This is the length of all orange lines above

  7. Investigation 1: Using technology plot the following: y = sinx y = 2sinx For each graph: • Find the maximum and minimum value • Find the period and amplitude Describe the effect of Ain the function y = Asinx • What is the amplitude of: y = 4sinx y = ⅔sinx

  8. 2 1  2 0 – 1 – 2 y = 2 sin x y = sin x

  9. 2 1  2 0 – 1 – 2 2 2 The period of each graph is the same, i.e. 2 But the amplitude of the new graph is 2 Y = A sinx- Vertical Stretch

  10. Investigation 2: Using technology plot the following: y = sinx y = sin2x For each graph: • Find the maximum and minimum value • Find the period and amplitude Describe the effect of Bin the function y = cosBx • What is the period of: y = cos4x y = cos¼x

  11. y = sin x 1 0 /2  2 – 1 y = sin 2x

  12. y = sin x 1 0 /2  2 – 1 y = sin 2x  Period is halved, and equals . Amplitudeis unchanged (1on both curves) Y = sinBx- Horizontal Stretch if |B| > 1 period is shorter 0 < B < 1 period is longer – stretched out

  13. Investigation 3: Using technology plot the following: y = sinx y = -2sinx, y = 2 sinx y = sin(-2x), y = sin2x Describe the effect of the negative in the trig functions. The negative means a reflection in the x-axis

  14. Investigation 4: y = sin(x) + 2 y = sin(x) – 1 • Calculate the equation of the principal axis y = sin(x) + D Vertical Translation if D is positive – shift up D is negative – shift down Equation of principal axis (new x axis): y = D

  15. The original curve is moved vertically up 2 units. Amplitude & period unchanged y = sin x + 2 y = sin x (“original”)

  16. Investigation 5: y = sin(x - 450) y = sin(x + 600) y = sin(x - C) Horizontal Translation if C is positive – shift curve to right if C is negative – shift curve to left

  17. IN GENERAL: y = AsinBx + D y = AsinB(x – C) + D To find: • Period = 360/B for degrees, 2π/B for radians • Principal axis y = D Affects Amplitude Affects Period Affects Principal Axis i.e Vertical Translation Affectshorizontal translation

  18. Graph Amplitude, A Period, T y= sin x 1 2 y= cos x y= 5sin x y= – 3 cos x y= sin 7x y= cos ¼x y= 3sin 2x y= 2 cos Complete the table 2 1 2 5 3 & is upside down 2 1 2/7 2/ ¼ i.e. 8 1 3 2/2 i.e.  2 2/ i.e. 6

  19. Example. Find the period and amplitude of y = 3sin x Write down: a = 3 and b =  Amplitude (A) is equal to 3. Period (T) is equal to 2/Bi.e. 2/ = 2

  20. Example. Find the period and amplitude of y = – 5 cos (x/6) Write down: a = 5 and B= /6 Noting that Amplitude (A) is equal to 5. Period (T) is equal to 2/(/6) i.e. 12

  21. Example. Find A, T and Band hence the equation of these graphs: 2 4 1. Goes through 0, so its a sine 2. Amplitude = 2. i.e. A= 2 T = 2 /B so 4 = 2 /B B = 2 /4  B = 0.5. 3. Period T = 4 . y= 2 sin 0.5x

  22. Page 273 Exercises 13D.1 13D.2

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