prove by mathematical induction,cos

Asked by keeruzzzzz | 11th Mar, 2020, 07:35: PM

Expert Answer:

To prove by mathematical induction: (cos theta + i sin theta)n = cos(n theta) + i sin (ntheta), where i = sqrt{-1}
F o r space n equals 1 comma
open parentheses cos theta space plus space i space sin theta close parentheses to the power of n equals open parentheses cos theta space plus space i space sin theta close parentheses
T h e r e f o r e comma space i t space i s space t r u e space t h a t space open parentheses cos theta space plus space i space sin theta close parentheses to the power of n equals open parentheses cos space n theta space plus space i space sin space n theta close parentheses space f o r space space n equals 1
L e t space u s space a s s u m e space t h a t space i t space i s space t r u e space f o r space n equals k
i. e. space open parentheses cos theta space plus space i space sin theta close parentheses to the power of k equals open parentheses cos space k theta space plus space i space sin space k theta close parentheses space... space left parenthesis i right parenthesis
T a k e space n equals k plus 1
open parentheses cos theta space plus space i space sin theta close parentheses to the power of k plus 1 end exponent
equals open parentheses cos theta space plus space i space sin theta close parentheses to the power of k open parentheses cos theta space plus space i space sin theta close parentheses space
equals open parentheses cos space k theta space plus space i space sin space k theta close parentheses open parentheses cos theta space plus space i space sin theta close parentheses space space... space F r o m space left parenthesis i right parenthesis
equals cos left parenthesis k theta right parenthesis space cos theta space plus space i space cos left parenthesis k theta right parenthesis space sin theta space plus space i space sin left parenthesis k theta right parenthesis space cos theta space minus space sin left parenthesis k theta right parenthesis space sin theta
equals cos left parenthesis k theta right parenthesis space cos theta space minus space sin left parenthesis k theta right parenthesis space sin theta space plus space i space open square brackets cos left parenthesis k theta right parenthesis space sin theta space plus space sin left parenthesis k theta right parenthesis space cos theta close square brackets
equals cos open parentheses k theta plus theta close parentheses space plus space i space sin open parentheses k theta plus theta close parentheses
equals cos open square brackets left parenthesis k plus 1 right parenthesis theta close square brackets plus i space sin open square brackets left parenthesis k plus 1 right parenthesis theta close square brackets
T h u s comma space open parentheses cos theta space plus space i space sin theta close parentheses to the power of k plus 1 end exponent equals cos open square brackets left parenthesis k plus 1 right parenthesis theta close square brackets plus i space sin open square brackets left parenthesis k plus 1 right parenthesis theta close square brackets
H e n c e comma space b y space m a t h e m a t i c a l space i n d u c t i o n space open parentheses cos theta space plus space i space sin theta close parentheses to the power of n space equals space open parentheses cos space n theta space plus space i space sin space n theta close parentheses space i s space t r u e space f o r space a l l space n element of straight natural numbers.

Answered by Renu Varma | 12th Mar, 2020, 12:45: PM