Complete expression for $3cos^{-1}x$
Complete expression for $3cos^{-1}x$
My Attempt
Let $a=cos^{-1}ximplies x=cos a$
$$
cos3a=4cos^3a-3cos a=4x^3-3x=sinbigg[sin^{-1}Big(4x^3-3xBig)bigg]\
3a=3cos^{-1}x=2npipmcos^{-1}Big(4x^3-3xBig)\
boxed{cos^{-1}Big(4x^3-3xBig)=2npipmbig(3cos^{-1}xbig)}
$$
$$
a=cos^{-1}xin [0,pi]implies3ain[0,3pi]
$$
Case 1: $3ain [0,pi]$
$$
color{darkblue}{3cos^{-1}x=cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin [0,pi]impliescos^{-1}xin [0,pi/3]implies xin [1/2,1]
$$
Case 2: $3ain (pi,2pi]$
$$
color{darkblue}{3cos^{-1}x=2pi-cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (pi,2pi]impliescos^{-1}xin (pi/3,2pi/3]implies xin [-1/2,1/2)
$$
Case 3: $3ain (2pi,3pi]$
$$
color{darkblue}{3cos^{-1}x=-2pi+cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (2pi,3pi]impliescos^{-1}xin (2pi/3,pi]implies xin [-1,-1/2)
$$
How do I confirm that my attempt provides a complete expression for $3cos^{-1}x$.
trigonometry inverse-function
asked Dec 6 at 21:20
ss1729
1,786722
add a comment |
Complete expression for $3cos^{-1}x$
My Attempt
Let $a=cos^{-1}ximplies x=cos a$
$$
cos3a=4cos^3a-3cos a=4x^3-3x=sinbigg[sin^{-1}Big(4x^3-3xBig)bigg]\
3a=3cos^{-1}x=2npipmcos^{-1}Big(4x^3-3xBig)\
boxed{cos^{-1}Big(4x^3-3xBig)=2npipmbig(3cos^{-1}xbig)}
$$
$$
a=cos^{-1}xin [0,pi]implies3ain[0,3pi]
$$
Case 1: $3ain [0,pi]$
$$
color{darkblue}{3cos^{-1}x=cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin [0,pi]impliescos^{-1}xin [0,pi/3]implies xin [1/2,1]
$$
Case 2: $3ain (pi,2pi]$
$$
color{darkblue}{3cos^{-1}x=2pi-cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (pi,2pi]impliescos^{-1}xin (pi/3,2pi/3]implies xin [-1/2,1/2)
$$
Case 3: $3ain (2pi,3pi]$
$$
color{darkblue}{3cos^{-1}x=-2pi+cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (2pi,3pi]impliescos^{-1}xin (2pi/3,pi]implies xin [-1,-1/2)
$$
How do I confirm that my attempt provides a complete expression for $3cos^{-1}x$.
trigonometry inverse-function
asked Dec 6 at 21:20
ss1729
1,786722
math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06
add a comment |
Complete expression for $3cos^{-1}x$
My Attempt
Let $a=cos^{-1}ximplies x=cos a$
$$
cos3a=4cos^3a-3cos a=4x^3-3x=sinbigg[sin^{-1}Big(4x^3-3xBig)bigg]\
3a=3cos^{-1}x=2npipmcos^{-1}Big(4x^3-3xBig)\
boxed{cos^{-1}Big(4x^3-3xBig)=2npipmbig(3cos^{-1}xbig)}
$$
$$
a=cos^{-1}xin [0,pi]implies3ain[0,3pi]
$$
Case 1: $3ain [0,pi]$
$$
color{darkblue}{3cos^{-1}x=cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin [0,pi]impliescos^{-1}xin [0,pi/3]implies xin [1/2,1]
$$
Case 2: $3ain (pi,2pi]$
$$
color{darkblue}{3cos^{-1}x=2pi-cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (pi,2pi]impliescos^{-1}xin (pi/3,2pi/3]implies xin [-1/2,1/2)
$$
Case 3: $3ain (2pi,3pi]$
$$
color{darkblue}{3cos^{-1}x=-2pi+cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (2pi,3pi]impliescos^{-1}xin (2pi/3,pi]implies xin [-1,-1/2)
$$
How do I confirm that my attempt provides a complete expression for $3cos^{-1}x$.
trigonometry inverse-function
asked Dec 6 at 21:20
ss1729
1,786722
Complete expression for $3cos^{-1}x$
My Attempt
Let $a=cos^{-1}ximplies x=cos a$
$$
cos3a=4cos^3a-3cos a=4x^3-3x=sinbigg[sin^{-1}Big(4x^3-3xBig)bigg]\
3a=3cos^{-1}x=2npipmcos^{-1}Big(4x^3-3xBig)\
boxed{cos^{-1}Big(4x^3-3xBig)=2npipmbig(3cos^{-1}xbig)}
$$
$$
a=cos^{-1}xin [0,pi]implies3ain[0,3pi]
$$
Case 1: $3ain [0,pi]$
$$
color{darkblue}{3cos^{-1}x=cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin [0,pi]impliescos^{-1}xin [0,pi/3]implies xin [1/2,1]
$$
Case 2: $3ain (pi,2pi]$
$$
color{darkblue}{3cos^{-1}x=2pi-cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (pi,2pi]impliescos^{-1}xin (pi/3,2pi/3]implies xin [-1/2,1/2)
$$
Case 3: $3ain (2pi,3pi]$
$$
color{darkblue}{3cos^{-1}x=-2pi+cos^{-1}Big(4x^3-3xBig)}\
3cos^{-1}xin (2pi,3pi]impliescos^{-1}xin (2pi/3,pi]implies xin [-1,-1/2)
$$
How do I confirm that my attempt provides a complete expression for $3cos^{-1}x$.
trigonometry inverse-function
trigonometry inverse-function
asked Dec 6 at 21:20
ss1729
1,786722
asked Dec 6 at 21:20
ss1729
1,786722
asked Dec 6 at 21:20
ss1729
1,786722
asked Dec 6 at 21:20
ss1729
1,786722
asked Dec 6 at 21:20
ss1729
1,786722
1,786722
math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06
add a comment |
math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06
math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06
add a comment |
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math.stackexchange.com/questions/1243735/…
– lab bhattacharjee
Dec 7 at 1:57
@labbhattacharjee the post u mentioned talks abt similar expression for $3sin^{-1}x$. Could u please comment on the case with $3cos^{-1}x$ ?
– ss1729
Dec 7 at 7:06