Forum
Forums
Questions and Answe...
Algebra
Inequalities
Inequalities with t...
 
Share:
Notifications
Clear all

[Solved] Inequalities with trigonometry

 
Inequalities
Last Post by Staff 4 years ago
1 Posts
1 Users
0 Reactions
503 Views
RSS
0
June 15, 2021 12:34 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

Find the value of m which statisfies the inequality

Add a comment
1 Answer
0
June 15, 2021 12:34 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small cos^2a+2msina-2m-2<0\Rightarrow sin^2a-2msina+2m+1>0.

Let   \small sina=t,  then \small -1\leq t\leq 1.

Assume \small f(t)=t^2-2mt+2m+1=(t-m)^2-m^2+2m+1>0, t\epsilon [-1,1].

(1)

If   \small m<-1, then \small f(t)_{min}=2+4m  at  \small t=-1.

Let \small 2+4m>0,   then  \small m>-\frac{1}{2}.   It is contradiction with \small m<-1.  Therefore   \small m>-\frac{1}{2}   should be rejected.

(2)

If \small -1\leq m\leq 1,  then \small f(t)_{min}=-m^2+2m+1  at  \small t=m.  

Let \small -m^2+2m+1>0,  that is  \small m^2-2m-1<0,  then \small 1-\sqrt{2}<m\leq 1.

(3)

If \small m>1,  then \small f(t)_{min}=2>0  at \small t=1.

After all \small m>1-\sqrt{2}.

Add a comment
Forum Jump:
  Previous Topic
Share: