Note that angle CEA = angle DCA + angle DAC and angle BDE = angle DCE + angle EDC, so angle DCA = angle CEA - angle DAC and angle DCE = angle EDC - angle CED.
Also, angle BDE = angle ABD + angle ADB = angle AEC + angle DAC, so angle DCE + angle DCA = (angle EDC - angle CED) + angle DCA.
Therefore, angle ACE = angle ACD + angle DCE = angle CEA, which implies that triangle ACE is isosceles.
Thanks MathCuber
Attn: asker
this is a good concept for you to work out quickly.
\(\sqrt{169}=+13\qquad \text{(never -13, this is by convention)}\\ \)
However if
\(y^2=169\)
then it is you that must introduce the square root sign,
which means that both the negative and the positive are answer.
\(y^2=169\\ y=\pm\sqrt169\\ y=+13\;\;\;\;or\;\;\;\;y=-13\)
If x - a is a factor of x^3 - 3*a*x^2 + 2*a^2*x + b, then find the value of b.
Wenn x - a ein Faktor von x ^ 3 - 3 * a * x ^ 2 + 2 * a ^ 2 * x + b ist, dann finde den Wert von b.
Hello Guest!
\(\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x + b}{ x-a}\\ =\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}+\frac{b}{x-a}\\ -\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}=\frac{b}{x-a}\\ \)
\(b=-x ^ 3 +3 a x ^ 2 -2 a ^ 2 x\)
!
asinus
I may not have done this the best way but this was my method:
1) Find BC
2) find angle BCP
3) Find angle ACQ
4) Find angfle CAQ
5) Now you have 3 angles and one side of triangle ACQ so you can find AQ
This is a teaching answer. Please do not override it.
MathCuber, if you have questions/problems then please ask.
By the Law of Cosines on triangle ABC, cos theta works out to 2/3.
Hi MathCuber,
I did this without any trig until the very end.
Just expand and simplify for starters.
Then compare what you have to the cosine rule and you should be able to work it out.
If you have questions then ask. Show or tell me what you have done though.
This is a teaching answer. Please noone interfer with a more full answer.
Asker guest:
Please reinstate the original address.
If questions are repeats they should always be linked with the original.
Thanks Alan :)
Taking the square root of 169, you get y = 13.
But we also have to remember that y can be negative, giving us that y = -13.
So our answers are y = 13, and y = -13.
There are 4 ways to color the red sides, then three ways to color the yellow sides, then two ways to color the green sides, then o one way to color the blue sides. Answer = 4*3*2*1 = 24.
Happy New Year! Alan
Sorry! Where are my manners? Happy Fucking New Year!
That is truly a complicated answer.
Thanks Alan. You are absolutely right. I just found out this about Wolfram: when your enter a set of numbers such as:112243, the first thing the computing engine does is to put them in "numerical order" and then calculate the permutations on them in numerical order. in other words, it treats:112234 exactly the same as:112243, hence the duplication in reversed groups of: 34 and 43, 35 and 53, 36 and 63, 45 and 54, 46 and 64, 56 and 65. And that is the difference of 1,080 =6 groups x 180
Wow! 212th decade already?
Hi guest, since the least common multiple of 12 and 6 is 12, there will be 12 days from now when she will go both kicking and swimming again on the same day
Split the integral into two parts, so you have int(x3cos(x/2)sqrt(4-x2))dx + (1/2)int(sqrt(4-x2))dx
The kernel of the first integral is odd, so when integrating from -2 to +2, the result is zero.
For the second integral make the substitution x = 2theta, say, and it is a simple matter to find the result as pi.
Yes, but if you look at the set of permutations for 112234, say, in WolframAlpha, you will see that it is identical to that for 112243.
wow do you have to be so rude on new years day
The least common multiple of 6 and 12 is 12.
So Helena will go kicking and swimming on the same day 12 days from now.
Instead of posting bullsh*t with \(\bf \pi\), demonstrate the solution...
Thanks Alan: I ran them on WolframAlpha and, in my experience, it NEVER lists duplicates! Please be good enough to look at this Wolfram page where I ran the 180 permutations of:112234:
https://www.wolframalpha.com/input/?i=permutations+of+%7B1%2C+1%2C+2%2C+2%2C+3%2C+4%7D
P.S. if you find it difficult to read, you may press the "plain text" button, which might be easier to read. Thanks.
The complex numbers are all different though
So I don't think this is correct. Not that I can do better or anything. Keep up the good work!
Using numerical integration techniques,
\(\int _{-2}^2\left(x^3\cos \left(\frac{x}{2}\right)+\frac{1}{2}\right)\sqrt{4-x^2}dx\) would be \(\pi\)
The four complex nubmers are \(\pm 5 \pm 4i\), so the area is 2*5*2*4 = 80.
The factors of 1024 would be 1, 2, 4, 8, 16, 32, 62, 128, 256, 512, and 1024 itself.
These numbers added up would give us 2047.
Thus, the answer to this question is 2047.
