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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: unknown id on March 09, 2008, 02:30:11 pm

Title: Inverse Trig
Post by: unknown id on March 09, 2008, 02:30:11 pm: Got stuck on this question. Any help would be appreciated:

Prove that $cos^{-1}(-x) = {\pi} - cos^{-1}(x)$
Title: Re: Inverse Trig
Post by: Mao on March 09, 2008, 02:43:21 pm: maybe try taking cos on both sides then expanding the RHS with compound angle formula??? :P
Title: Re: Inverse Trig
Post by: unknown id on March 09, 2008, 02:45:20 pm: i could have done that but i didn't want to touch the right-hand side when proving
Title: Re: Inverse Trig
Post by: Mao on March 09, 2008, 02:54:12 pm: why not?

$cos^{-1}(-x)=\pi-cos^{-1}(x)$

$cos \left(cos^{-1}(-x) \right)=cos \left( \pi-cos^{-1}(x) \right)$

$-x=cos(\pi) \cdot x + sin(\pi) \cdot sin\left( cos^{-1}(x) \right)$

$-x=-x+0$

QED :D

you can also use the symmetry of cos:

$x=cos(\theta)$

$\Rightarrow \theta=cos^{-1}(x)$

we also know that $cos(\pi-\theta)=-cos(\theta)$

$cos(\pi-\theta)=-x$

$\Rightarrow cos^{-1}(-x)=\pi - \theta$

$\Rightarrow cos^{-1}(-x)=\pi-cos^{-1}(x)$

QED

i think this would be your preferred method? :P
Title: Re: Inverse Trig
Post by: unknown id on March 09, 2008, 02:58:45 pm: i thought that whenever we have to prove that the left side equals the right side, only the left side should be used in our calculations
Title: Re: Inverse Trig
Post by: Mao on March 09, 2008, 03:04:25 pm: Quote from: unknown id on March 09, 2008, 02:58:45 pm
i thought that whenever we have to prove that the left side equals the right side, only the left side should be used in our calculations
however (more realistically), if you can arrive at 0=0 by any means, it'll be eureka! regardless

i have since edited my post to include a second proof with your preferred "only involve the left side" :P
even though the beginnning step is not $cos^{-1}(-x)$
Title: Re: Inverse Trig
Post by: unknown id on March 09, 2008, 03:05:08 pm: thanks mao, the second method does seem more preferable :D
Title: Re: Inverse Trig
Post by: gfb on March 09, 2008, 03:14:58 pm: Good work Mao ;).