MHB Katlynsbirds' question at Yahoo Answers regarding inverse trigonometric identity

AI Thread Summary
The identity to prove is cot inverse of x equals sin inverse of 1 over the square root of 1 plus x squared. By letting theta equal cot inverse of x, it follows that x equals cot(theta). A diagram illustrates that sin(theta) equals 1 over the square root of 1 plus x squared, leading to the conclusion that theta equals sin inverse of 1 over the square root of 1 plus x squared. This confirms the identity as required. The discussion encourages further trigonometry problems to be shared in the forum.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Prove the identity, pre calc!?

cot inverse= sin inverse of 1/sqr of 1+x^2

Here is a link to the question:

Prove the identity, pre calc!? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Re: katlynsbirds' question at Yahoo! Answers regarding inverse trignometric identity

Hello katlynsbirds,

We are given to prove:

$$\cot^{-1}(x)=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

Let's let $$\theta=\cot^{-1}(x)\,\therefore\,x=\cot(\theta)$$, and now please refer to this diagram:

https://www.physicsforums.com/attachments/765._xfImport

We see that $$\cot(\theta)=\frac{x}{1}=x$$ and we can also see that:

$$\sin(\theta)=\frac{1}{\sqrt{1+x^2}}\,\therefore\, \theta=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

and so we may conclude:

$$\theta=\cot^{-1}(x)=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

Shown as desired.

To katlynsbirds and any other guests viewing this topic I invite and encourage you to post other trigonometry problems here in our http://www.mathhelpboards.com/f12/ forum.

Best Regards,

Mark.
 

Attachments

  • katlynsbirds.jpg
    katlynsbirds.jpg
    3.5 KB · Views: 88
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top