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 (b) Max = 3 when x = 0 or 360 (c) Min = -3 when x = 180 (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) So 3cosx° + 4sinx° = 5cos(x – 53.1)° 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 why 3cosx° + 4sinx° = 5cos(x – 53.1)° The Wave Function kcos(x - )° Note: any function in the form acosx° + bsinx° kcos(x - )° can be expressed in the form where a = kcos° and MUST LEARN !! b = ksin° . By considering the trig addition formulae and comparing coefficients we can prove the above as follows…. 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 also and a2 + b2 = = = = b/ a b = ksin° (kcos°)2 + (ksin°)2 k2cos2° + k2sin2° k2(cos2° + sin2°) k2 = ksin° = tan° kcos° common factor! cos2° + sin2° = 1 So k2 = a2 + b2 and tan = b/a The Wave Function ksin(x + )° Note: any function in the form asinx° + bcosx° ksin(x + )° can be expressed in the form where a = kcos° and MUST LEARN !! b = ksin° . By considering the trig addition formulae and comparing coefficients we can prove the above as follows…. 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 also and a2 + b2 = = = = b/ a b = ksin° (kcos°)2 + (ksin°)2 k2cos2° + k2sin2° k2(cos2° + sin2°) k2 = ksin° = tan° kcos° just like before common factor! cos2° + sin2° = 1 So k2 = a2 + b2 and tan = b/a SUMMARY acosx° + bsinx° = kcos(x - )° asinx° + bcosx° = ksin(x + )° In both cases.. a = kcos° and b = ksin° and tan ° = b/a which leads to k2 = a2 + b2