This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.
Arccos + Arcsin
Let y = arccos(x)
This means that x = cos(y)
This gives us this triangle, as cosine is adjacent ÷ hypotenuse.
This means that x = cos(y)
This gives us this triangle, as cosine is adjacent ÷ hypotenuse.
Call the angle at the top of the triangle z
Sine is opposite ÷ hypotenuse, so sin(z) = x
So z = arcsin(x)
Angles in triangle add to π, so z = π/2 - y
Hence, arcsin(x) = π/2 - y
Therefore arccos(x) + arcsin(x) = y + π/2 - y = π/2
Sine is opposite ÷ hypotenuse, so sin(z) = x
So z = arcsin(x)
Angles in triangle add to π, so z = π/2 - y
Hence, arcsin(x) = π/2 - y
Therefore arccos(x) + arcsin(x) = y + π/2 - y = π/2
Extension
What are the values of arcsec(x) + arccosec(x) and arctan(x) + arccot(x)?
i i
It is notable that this is a real number.
Extension
There are other values of i to the power of i. What are they?
xxxxxx...
2 = xxxxxx... = x2.
So x = √2.
So x = √2.
Extension
If xxxxxx... is equal to four, what is the value of x?
4 and 5 digit numbers
Adam must have removed the final digit of his number before adding or the final digit of the sum would have been an even number. If his original number was ABCDE then, using this, we have:
52713 = ABCDE + ABCD = 11 × ABCD + E.
52713 ÷ 11 = 4792 remainder 1 so Adam's original number was 47921 which has a digit sum of 23.
I don't think there are any other values of i^i.
ReplyDeleteI was assuming that you had the R x [0,2pi] definition of (r, theta) to define C. I guess you can have e^(pi/2 + k*2*pi) solutions if you define it on the non-bijective R x R definition of (r, theta).
ReplyDeleteI don't like your triangle proof - what happens when y is outside (0, Π\2) you can't have negative sides to a triangle!
ReplyDelete