Solve the problem that involves ##\cos^{-1} x + \cos^{-1}y##

AI Thread Summary
The discussion focuses on solving the equation involving the sum of inverse cosines, specifically ##\cos^{-1} x + \cos^{-1} y##. A right-angled triangle approach is utilized, defining angles C and A corresponding to x and y, respectively, leading to the relationship A + C = π/2. The key derivation shows that the expression simplifies to ##\cos^{-1}(0) = π/2##, indicating that the left-hand side equals the right-hand side. The conversation highlights that the assumption of x and y being on the unit circle is unnecessary for the solution. The final conclusion reaffirms the trigonometric identity used in the calculations.
chwala
Gold Member
Messages
2,827
Reaction score
415
Homework Statement
See attached.
Relevant Equations
Trigonometry
1694237381792.png
In my approach (using a right angled triangle) i let,

##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##

and

##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##

Also, ##A+C = \dfrac{π}{2}##

and ##\cos \dfrac{π}{2}= 0##

##xy - \sqrt{(y^2) ⋅(x^2)}=xy-xy=0##

It follows that,

##\cos^{-1} [xy - \sqrt{(1-x^2)(1-y^2)}]= \cos^{-1}[ xy - \sqrt{(y^2) ⋅(x^2)}]##

##=\cos^{-1} (xy - \sqrt{y^2}⋅ \sqrt{x^2})=\cos^{-1} (xy-xy)=\cos^{-1} (0)= \dfrac{π}{2}##
 
Physics news on Phys.org
Cos of the LHS =
\cos(\cos^{-1}x+\cos^{-1}y)=\cos(\cos^{-1}x)\cos(\cos^{-1}y)-\sin(\cos^{-1}x)\sin(\cos^{-1}y)
=xy-\sqrt{1-x^2}\sqrt{1-y^2}
 
  • Like
Likes chwala and SammyS
chwala said:
Homework Statement: See attached.
Relevant Equations: Trigonometry

View attachment 331694In my approach (using a right angled triangle) i let,

##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##

and

##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##

Also, ##A+C = \dfrac{π}{2}##
You seem to be assuming that ##\displaystyle x^2+y^2=1 \, , \ ## as if ##(x,\, y) \ ## is an ordered pair on the unit circle. That assumption is not necessary.

If ## \displaystyle \cos^{-1} x = C \, , \ ## then ##\displaystyle \cos C = x \ ## and ##\displaystyle \sin C = \sqrt{ 1-x^2} \ . \ ##

etc.

See Post #2 by @anuttarasammyak .
 
Last edited:
anuttarasammyak said:
Cos of the LHS =
\cos(\cos^{-1}x+\cos^{-1}y)=\cos(\cos^{-1}x)\cos(\cos^{-1}y)-\sin(\cos^{-1}x)\sin(\cos^{-1}y)
=xy-\sqrt{1-x^2}\sqrt{1-y^2}
ok i see the logic now

##\cos(A+B)=\cos A \cos B - \sin A \sin B##.

Cheers Man!
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
Back
Top