
Tuesday, October 30, 2018 
Wednesday, April 24, 2019 10:33:18 PM 
382 [0.04% of all post / 2.15 posts per day] 

Thank you very much, DS.
My English is probably at CEFR A1 or A2 level.

thar wrote:That isn't a formula, is it, so much as an equation?
I just found this:
Machin's formula (for which the derivation is straightforward) is:
π/4 = 4 arctan 1/5 – arctan 1/239.
Wikipedia, John Machin
My English is probably at CEFR A1 or A2 level.

Blodybeef wrote:Quote:Is for really necessary? Is the following sentence correct? It sounds better with "for" However, I would personally construct this sentence like this : "The price was raised seven times since 2012."
I suppose the "a total of seven times" in my sentence is adverbial.
My English is probably at CEFR A1 or A2 level.

Koh Elaine wrote:“I received the cheques from Najib for a total of three times.
Drag0nspeaker wrote:Hi! It seems correct  except that it's just long and "wordy". Is for really necessary? Is the following sentence correct?
The price has been raised a total of seven times since 2012. (Without for.) (my original sentence)
My English is probably at CEFR A1 or A2 level.

Thank you all,
thar wrote:That isn't a formula, is it, so much as an equation? Serge Lang, the author of the acclaimed algebra text Algebra, called it a formula.
Here's my proof of the formula:
α := arctan 1/5. tan α = 1/5. tan 2α = 5/12. tan 4α = 120/119. tan (4α  π/4) = 1/239. α < π/6 because 1/5 < tan π/6. Thus, – π/2 < 4α  π/4 < π/2. Hence, 4α  π/4 = arctan 1/239. That is, π/4 = 4 arctan 1/5 – arctan 1/239.
We can use this formula to show: π ≈ 3.1415926.
My English is probably at CEFR A1 or A2 level.

thar wrote:"sine to the minus one" Thank very much, thar.
You might have taken sin⁻¹ x to mean (sin x)⁻¹. I mean:
sin⁻¹ x = arcsin x, not sin⁻¹ x = (sin x)⁻¹.
Some Japanese math teachers pronounce sin⁻¹ x as "inverse sine x".
My English is probably at CEFR A1 or A2 level.

(1) arcsin x (2) arccos x (3) arctan x
(4) sin⁻¹ x (5) cos⁻¹ x (6) tan⁻¹ x
How would you pronounce these expressions?
Example:
Prove the formula π/4 = 4 arctan 1/5 – arctan 1/239.
My English is probably at CEFR A1 or A2 level.

Thanks a lot, thar.
My English is probably at CEFR A1 or A2 level.

Thanks a lot, DS.
My English is probably at CEFR A1 or A2 level.

(1) Look at the sleeping baby. [from a grammar book for Japanese learners of English]
(2) Look at the baby sleeping.
Is (1) natural? I think (2) is better.
My English is probably at CEFR A1 or A2 level.

