trigonometric inequalities


by junaidnawaz
Tags: arccos, inequalities, relational operators, trigonometry
junaidnawaz
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#1
Feb11-11, 03:29 AM
P: 6
Please help me to confirm, weather the following step is correct

[tex] |\gamma| \leq \cos (\beta) [/tex]
[tex] \arccos (|\gamma|) \leq \beta [/tex]

does taking the arccos() on both sides of equation changes the relational operator??
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tiny-tim
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#2
Feb11-11, 03:42 AM
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hi junaidnawaz! welcome to pf!

(have a beta: β and a gamma: γ and a ≤ )

arccos is defined as being in [0,π)

so long as β is also in [0,π), your equations are the same (because cos is monotone in that region, and therefore so is arccos)
junaidnawaz
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#3
Feb11-11, 04:08 AM
P: 6
Thx v much for your reply.

in my case, the range of parameters is as,
[tex] 0 \leq \beta \leq \pi /2 [/tex]
[tex]-1 \leq \gamma \leq +1 [/tex]

by taking arccos() on both-sides, would it change the operator (from [tex] \leq[/tex] to [tex] \geq[/tex] ) or would it remain same ??

tiny-tim
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Feb11-11, 04:21 AM
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trigonometric inequalities


Quote Quote by junaidnawaz View Post
by taking arccos() on both-sides, would it change the operator (from [tex] \leq[/tex] to [tex] \geq[/tex] ) or would it remain same ??
oh, i missed that!

yes, cos is decreasing, so the ≤ changes to ≥

(but, eg, sin is increasing, so the ≤ would stay the same )
junaidnawaz
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#5
Feb11-11, 05:08 AM
P: 6
Thank you.

if
[tex] |\gamma| \leq \cos( \beta ) [/tex]

then

[tex] x \leq \beta[/tex]

can i find "x", by keeping the RHS fixed to [tex] \beta[/tex]

is this possible to find x ?? by keeping RHS and relation operator the same ??
tiny-tim
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#6
Feb11-11, 05:24 AM
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sorry, i don't understand

isn't x just |γ| ?
junaidnawaz
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#7
Feb11-11, 05:32 AM
P: 6
[tex] |\gamma| \leq \cos ( \beta ) [/tex]

When I take arccos() on both sides, it becomes

[tex] \arccos( |\gamma| ) \geq \beta [/tex]

however, i want to keep [tex] \beta [/tex] on right side, and i want to keep the relational operator as [tex] \leq [/tex], i.e.,

[tex] x \leq \beta [/tex]

what would be x ??
junaidnawaz
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#8
Feb11-11, 05:54 AM
P: 6
I wounder if its not a stupid question .... :P
tiny-tim
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Feb11-11, 06:08 AM
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Quote Quote by junaidnawaz View Post
however, i want to keep [tex] \beta [/tex] on right side, and i want to keep the relational operator as [tex] \leq [/tex], i.e.,

[tex] x \leq \beta [/tex]
that's not possible (unless you replace β by some decreasing function of β, such as 1/β or -β)
junaidnawaz
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#10
Feb11-11, 07:15 AM
P: 6
Thank you :)


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