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WAVE FUNCTIONS

WAVE FUNCTIONS. Suppose we compare the following 3 graphs. (i) y = 4sinx °. (ii) y = 3cosx °. (iii) y = 3cosx ° + 4sinx °. (i) y = 4sinx °. Main Features. (a) Wave shape. (b) Max = 4 when x = 90. (c) Min = -4 when x = 270. (ii) y = 3cosx °. Main Features. (a) Roller-coaster shape.

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WAVE FUNCTIONS

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  1. WAVE FUNCTIONS Suppose we compare the following 3 graphs (i) y = 4sinx° (ii) y = 3cosx° (iii) y = 3cosx° + 4sinx°

  2. (i) y = 4sinx° Main Features (a) Wave shape (b) Max = 4 when x = 90 (c) Min = -4 when x = 270

  3. (ii) y = 3cosx° Main Features (a) Roller-coaster shape (b) Max = 3 when x = 0 or 360 (c) Min = -3 when x = 180

  4. (iii) y = 3cosx° + 4sinx° Main Features (a) Roller-coaster shape but to right of Y-axis  cos(x - …) (b) Max = 5 when x  50 5cos(x – 50)° (c) Min = -5 when x  230 (actual values are 53.1 & 233.1) So3cosx° + 4sinx° = 5cos(x – 53.1)°

  5. By converting from a mixture of sinx & cosx to just cos(…..) or possibly sin(…) the function is easier to deal with. The function that uses just a single trig ratio – rather than a mixture of sin & cos – is called a wave function. We now look at why3cosx° + 4sinx° = 5cos(x – 53.1)°

  6. The Wave Function kcos(x - )° Note: any function in the form acosx° + bsinx° MUST LEARN !! can be expressed in the form kcos(x - )° where a = kcos° and b = ksin° . By considering the trig addition formulae and comparing coefficients we can prove the above as follows….

  7. Proof Let acosx° + bsinx° = kcos(x - )° = k(cosx°cos° + sinx°sin°) = kcosx°cos° + ksinx°sin° = (kcos°)cosx° + (ksin°)sinx° Comparing coefficients we get a = kcos° and b = ksin° also a2 + b2 = (kcos°)2 + (ksin°)2 = k2cos2° + k2sin2° common factor! = k2(cos2° + sin2°) cos2° + sin2° = 1 = k2 ksin° So k2 = a2 + b2 and tan = b/a and b/a = = tan° kcos°

  8. The Wave Function ksin(x + )° Note: any function in the form asinx° + bcosx° MUST LEARN !! can be expressed in the form ksin(x + )° where a = kcos° and b = ksin° . By considering the trig addition formulae and comparing coefficients we can prove the above as follows….

  9. Proof Let asinx° + bcosx° = ksin(x + )° = k(sinx°cos° + cosx°sin°) = ksinx°cos° + kcosx°sin° = (kcos°)sinx° + (ksin°)cosx° Comparing coefficients we get a = kcos° and b = ksin° just like before also a2 + b2 = (kcos°)2 + (ksin°)2 = k2cos2° + k2sin2° common factor! = k2(cos2° + sin2°) cos2° + sin2° = 1 = k2 ksin° So k2 = a2 + b2 and tan = b/a and b/a = = tan° kcos°

  10. SUMMARY acosx° + bsinx° = kcos(x - )° asinx° + bcosx° = ksin(x + )° In both cases.. a = kcos° and b = ksin° which leads to k2 = a2 + b2 and tan ° = b/a

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