Can I use a calculator?
A B = = tan − 1 ( 2 1 ) + tan − 1 ( 3 1 ) + tan − 1 ( 4 1 ) 2 π
Without resorting to tables or calculator, determine which of these numbers above is larger.
All credit for this problem goes to Pi Han Goh .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
3 solutions
My last comment, if it posted (my phone is acting really weird lately, sorry), should say 2 tan − 1 ( ( 3 1 ) . And also, the first sentence should state that it's increasing at a decreasing rate. but, the difference between 1/2 and 3 1 is much smaller than that of 3 1 and 2 1
It is not that much greater than 135 degrees, so ideally there should be a better solution.
It is not greater than 135 degrees it is just a little over 90 degrees
Log in to reply
He added tan − 1 1 = 4 5 ∘ to the count. Look at the image.
Just it's way bigger...😂😂😂
Solution credit to Pi Han Goh
This question was inspired by Spiral of Theodorus .
Draw the first four triangles as shown above. The total angle subtended from the first triangle to the fourth triangle is n = 1 ∑ 4 tan − 1 ( n 1 ) .
Using a protractor, we can see that it's larger than 1 3 5 ∘ = 4 3 π .
Then, n = 1 ∑ 4 tan − 1 ( n 1 ) > 4 3 π .
Because tan − 1 ( 1 1 ) = 4 π , then n = 2 ∑ 4 tan − 1 ( n 1 ) > 2 π or simply A > B .