Expression for $cos^{-1}xpmcos^{-1}y$












1












$begingroup$


As mentioned in Proof for the formula of sum of arcsine functions $arcsin x+arcsin y$ for $sin^{-1}x+sin^{-1}y$



$$
sin^{-1}x+sin^{-1}y=
begin{cases}
sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 le 1 \
pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, 0< x,y le 1\
-pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, -1< x,y le 0
end{cases}
$$
Can we have similar expression for $cos^{-1}xpmcos^{-1}y$ ?






trigonometry inverse-function







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$endgroup$








  • 4




    $begingroup$
    I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
    $endgroup$
    – user7530
    Dec 5 '18 at 23:17












  • $begingroup$
    Also math.stackexchange.com/questions/1224415/…
    $endgroup$
    – lab bhattacharjee
    Dec 6 '18 at 6:47
















1












$begingroup$


As mentioned in Proof for the formula of sum of arcsine functions $arcsin x+arcsin y$ for $sin^{-1}x+sin^{-1}y$



$$
sin^{-1}x+sin^{-1}y=
begin{cases}
sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 le 1 \
pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, 0< x,y le 1\
-pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, -1< x,y le 0
end{cases}
$$
Can we have similar expression for $cos^{-1}xpmcos^{-1}y$ ?






trigonometry inverse-function







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
    $endgroup$
    – user7530
    Dec 5 '18 at 23:17












  • $begingroup$
    Also math.stackexchange.com/questions/1224415/…
    $endgroup$
    – lab bhattacharjee
    Dec 6 '18 at 6:47














1












1








1





$begingroup$


As mentioned in Proof for the formula of sum of arcsine functions $arcsin x+arcsin y$ for $sin^{-1}x+sin^{-1}y$



$$
sin^{-1}x+sin^{-1}y=
begin{cases}
sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 le 1 \
pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, 0< x,y le 1\
-pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, -1< x,y le 0
end{cases}
$$
Can we have similar expression for $cos^{-1}xpmcos^{-1}y$ ?






trigonometry inverse-function







share|cite|improve this question











$endgroup$




As mentioned in Proof for the formula of sum of arcsine functions $arcsin x+arcsin y$ for $sin^{-1}x+sin^{-1}y$



$$
sin^{-1}x+sin^{-1}y=
begin{cases}
sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 le 1 \
pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, 0< x,y le 1\
-pi - sin^{-1}( xsqrt{1-y^2} + ysqrt{1-x^2}) ;;;x^2+y^2 > 1, -1< x,y le 0
end{cases}
$$
Can we have similar expression for $cos^{-1}xpmcos^{-1}y$ ?






trigonometry inverse-function




trigonometry inverse-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 6 '18 at 6:40







ss1729

















asked Dec 5 '18 at 23:03









ss1729ss1729

1,9021723




1,9021723








  • 4




    $begingroup$
    I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
    $endgroup$
    – user7530
    Dec 5 '18 at 23:17












  • $begingroup$
    Also math.stackexchange.com/questions/1224415/…
    $endgroup$
    – lab bhattacharjee
    Dec 6 '18 at 6:47














  • 4




    $begingroup$
    I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
    $endgroup$
    – user7530
    Dec 5 '18 at 23:17












  • $begingroup$
    Also math.stackexchange.com/questions/1224415/…
    $endgroup$
    – lab bhattacharjee
    Dec 6 '18 at 6:47








4




4




$begingroup$
I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
$endgroup$
– user7530
Dec 5 '18 at 23:17






$begingroup$
I mean $arccos(x) = pi/2 - arcsin(x)$ which you can surely use to reformulate your expressions in terms of $arccos$.
$endgroup$
– user7530
Dec 5 '18 at 23:17














$begingroup$
Also math.stackexchange.com/questions/1224415/…
$endgroup$
– lab bhattacharjee
Dec 6 '18 at 6:47




$begingroup$
Also math.stackexchange.com/questions/1224415/…
$endgroup$
– lab bhattacharjee
Dec 6 '18 at 6:47










1 Answer
1






active

oldest

votes


















3












$begingroup$

Alternatively, to get rid of all those cases, we can write:
$$sin^{-1}xpmsin^{-1}y=text{atan2}(xsqrt{1-y^2}pm ysqrt{1-x^2}, sqrt{1-x^2}sqrt{1-y^2}mp xy)$$
and:
$$cos^{-1}xpmcos^{-1}y=text{atan2}(ysqrt{1-x^2}pm xsqrt{1-y^2}, xympsqrt{1-x^2}sqrt{1-y^2})$$



See atan2.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanx. but the cases are really important and my doubt is there !
    $endgroup$
    – ss1729
    Dec 6 '18 at 0:55










  • $begingroup$
    How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
    $endgroup$
    – I like Serena
    Dec 6 '18 at 8:29













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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









3












$begingroup$

Alternatively, to get rid of all those cases, we can write:
$$sin^{-1}xpmsin^{-1}y=text{atan2}(xsqrt{1-y^2}pm ysqrt{1-x^2}, sqrt{1-x^2}sqrt{1-y^2}mp xy)$$
and:
$$cos^{-1}xpmcos^{-1}y=text{atan2}(ysqrt{1-x^2}pm xsqrt{1-y^2}, xympsqrt{1-x^2}sqrt{1-y^2})$$



See atan2.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanx. but the cases are really important and my doubt is there !
    $endgroup$
    – ss1729
    Dec 6 '18 at 0:55










  • $begingroup$
    How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
    $endgroup$
    – I like Serena
    Dec 6 '18 at 8:29


















3












$begingroup$

Alternatively, to get rid of all those cases, we can write:
$$sin^{-1}xpmsin^{-1}y=text{atan2}(xsqrt{1-y^2}pm ysqrt{1-x^2}, sqrt{1-x^2}sqrt{1-y^2}mp xy)$$
and:
$$cos^{-1}xpmcos^{-1}y=text{atan2}(ysqrt{1-x^2}pm xsqrt{1-y^2}, xympsqrt{1-x^2}sqrt{1-y^2})$$



See atan2.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanx. but the cases are really important and my doubt is there !
    $endgroup$
    – ss1729
    Dec 6 '18 at 0:55










  • $begingroup$
    How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
    $endgroup$
    – I like Serena
    Dec 6 '18 at 8:29
















3












3








3





$begingroup$

Alternatively, to get rid of all those cases, we can write:
$$sin^{-1}xpmsin^{-1}y=text{atan2}(xsqrt{1-y^2}pm ysqrt{1-x^2}, sqrt{1-x^2}sqrt{1-y^2}mp xy)$$
and:
$$cos^{-1}xpmcos^{-1}y=text{atan2}(ysqrt{1-x^2}pm xsqrt{1-y^2}, xympsqrt{1-x^2}sqrt{1-y^2})$$



See atan2.






share|cite|improve this answer









$endgroup$



Alternatively, to get rid of all those cases, we can write:
$$sin^{-1}xpmsin^{-1}y=text{atan2}(xsqrt{1-y^2}pm ysqrt{1-x^2}, sqrt{1-x^2}sqrt{1-y^2}mp xy)$$
and:
$$cos^{-1}xpmcos^{-1}y=text{atan2}(ysqrt{1-x^2}pm xsqrt{1-y^2}, xympsqrt{1-x^2}sqrt{1-y^2})$$



See atan2.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 6 '18 at 0:25









I like SerenaI like Serena

4,1421721




4,1421721












  • $begingroup$
    thanx. but the cases are really important and my doubt is there !
    $endgroup$
    – ss1729
    Dec 6 '18 at 0:55










  • $begingroup$
    How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
    $endgroup$
    – I like Serena
    Dec 6 '18 at 8:29




















  • $begingroup$
    thanx. but the cases are really important and my doubt is there !
    $endgroup$
    – ss1729
    Dec 6 '18 at 0:55










  • $begingroup$
    How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
    $endgroup$
    – I like Serena
    Dec 6 '18 at 8:29


















$begingroup$
thanx. but the cases are really important and my doubt is there !
$endgroup$
– ss1729
Dec 6 '18 at 0:55




$begingroup$
thanx. but the cases are really important and my doubt is there !
$endgroup$
– ss1729
Dec 6 '18 at 0:55












$begingroup$
How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
$endgroup$
– I like Serena
Dec 6 '18 at 8:29






$begingroup$
How are the cases important, and which doubt do you have @ss1729? Note that we have a formula that transforms an angle uniquely into x,y. Arcsin and arccos don't do the reverse but atan2 does.
$endgroup$
– I like Serena
Dec 6 '18 at 8:29




















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