S5_Unit_3_Outcome_4 - www.mathsrevision.com Higher Outcome...

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Unformatted text preview: www.mathsrevision.com Higher Outcome 4 Higher Unit 3 www.mathsrevision.com www.mathsrevision.com What is a Wave Function Connection with Trig Identities Earlier Maximum and Minimum Values Exam Type Questions Solving Equations involving the Wave Function www.mathsrevision .com Higher Outcome 4 The Wave Func tio n cos sin cos( ) sin( ). Ex pr e s s ing in t h e f o r m + or a x b x k x k x He a rt b e a t Ele c tric Ma ny wa ve s ha pe s , whe the r o c c urring a s s o und, lig ht, wa te r o r e le c tric a l wa ve s , c a n b e de s c rib e d m a the m a tic a lly a s a c o m b ina tio n o f s ine a nd c o s ine wa ve s . S pe c trum Ana lys is www.mathsrevision .com Higher Outcome 4 Y The Wave Function General shape for y = sinx + cosx 1.Like y = sin(x) shifted left 2.Like y = cosx shifted right 3.Vertical height different y = sin(x) y = cos(x) y = sin(x)+cos(x) www.mathsrevision .com Higher Outcome 4 Whenever a function is formed by adding cosine and sine functions the result can be expressed as a related cosine or sine function. In general: cos sin cos( ) cos sin sin( ) + = + = or a x b x k x a x b x k x , , and are constants a b k Given and we can caculate and a b k With these constants the expressions on the right hand sides = those on the left hand side FOR ALL VALUES OF x The Wave Function www.mathsrevision .com Higher Outcome 4 Worked Example: 4cos 3sin cos( ) 360 +- Write in the form , where o o x x k x 4cos 3sin cos( ) + =- o x x k x cos( ) cos cos sin sin x x x - = + Re m e m b e r !!!!! ( 29 cos cos sin sin = + k x x Re-arrange 4cos 3sin cos cos sin sin + = + x x k x k x The left and right hand sides must be equal for all values of x. So, the coefficients of cos x and sin x must be equal : cos 4 sin 3 = = k k A pair of simultaneous equations to be solved The Wave Function www.mathsrevision .com Higher Outcome 4 cos 4 sin 3 = = k k 2 2 2 2 2 2 cos sin 4 3 + = + k k ( 29 2 2 2 cos sin 16 9 + = + k 2 25 = k 5 = k 1 sin 3 tan tan cos 4 - = = o =36.9 The Wave Function Find tan ratio note: sin(+) and cos(+) Square and add www.mathsrevision .com Higher Outcome 4 cos 4 sin 3 = = k k 5 = k 36.9 = o is in the first quadrant 4cos 3sin cos( ) + =- o x x k x 4cos 3sin 5cos( 36.9) + =- o x x x The Wave Function C A S T o 180 o 270 o 90 o 3 2 2 Note: sin(+) and cos(+) www.mathsrevision .com Higher Outcome 4 The Wave Function Example cos 3sin cos( ) 2 x x R x - + Write in the form , where...
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S5_Unit_3_Outcome_4 - www.mathsrevision.com Higher Outcome...

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