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estimation for sin function

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0

$begingroup$

We have that $$-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)=sqrta_n^2+b_n^2sin(sqrtlambda_nt+theta_n)$$ for some $theta_nin R$

Then $$|-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)|leqlambda_nsqrta_n^2+b_n^2 |t-t_0|$$

for every $t_0$ belonging to $t_0mid sin(lambda_nt_0+theta_n)=0$.

The question is: how we can get this inequality,and if there is some rules on the difference between $sin$ function that can give this. (I mean if $|sin(lambda_nt+theta_n)-sin(lambda_nt_0+theta_n)|leq lambda_n|t-t_0|$).

calculus algebra-precalculus trigonometry inequality

share|cite|improve this question

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

$endgroup$

add a comment | 

0

$begingroup$

We have that $$-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)=sqrta_n^2+b_n^2sin(sqrtlambda_nt+theta_n)$$ for some $theta_nin R$

Then $$|-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)|leqlambda_nsqrta_n^2+b_n^2 |t-t_0|$$

for every $t_0$ belonging to $t_0mid sin(lambda_nt_0+theta_n)=0$.

The question is: how we can get this inequality,and if there is some rules on the difference between $sin$ function that can give this. (I mean if $|sin(lambda_nt+theta_n)-sin(lambda_nt_0+theta_n)|leq lambda_n|t-t_0|$).

calculus algebra-precalculus trigonometry inequality

share|cite|improve this question

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

$endgroup$

add a comment | 

$begingroup$

We have that $$-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)=sqrta_n^2+b_n^2sin(sqrtlambda_nt+theta_n)$$ for some $theta_nin R$

Then $$|-a_nsin(sqrtlambda_nt)+b_ncos(sqrtlambda_nt)|leqlambda_nsqrta_n^2+b_n^2 |t-t_0|$$

for every $t_0$ belonging to $t_0mid sin(lambda_nt_0+theta_n)=0$.

The question is: how we can get this inequality,and if there is some rules on the difference between $sin$ function that can give this. (I mean if $|sin(lambda_nt+theta_n)-sin(lambda_nt_0+theta_n)|leq lambda_n|t-t_0|$).

calculus algebra-precalculus trigonometry inequality

share|cite|improve this question

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

$endgroup$

share|cite|improve this question

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

share|cite|improve this question

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

edited Mar 29 at 9:41

Dylan

14.2k31127

edited Mar 29 at 9:41

Dylan

14.2k31127

asked Mar 28 at 19:42

i.issai.issa

143

asked Mar 28 at 19:42

i.issai.issa

143