MaxWong

$P = 20 - \dfrac{1}{5}Q\\ 5P = 100 - Q\\ Q = 100 - 5P$

Given: $\sin a = -\dfrac{21}{29},\dfrac{3\pi}{2}<a<2\pi\\ \tan b = -\dfrac{24}{7},\dfrac{\pi}{2}<b<\pi$

Note that: $\boxed{\color{BurntOrange}{\cos(a + b) = \cos a \cos b - \sin a \sin b}}$

So that we have to find cos a, cos b, sin b because the value of sin a is given.

A few formulae: $\cos(\arcsin x) = \sqrt{1-x^2}\\ \cos(\arctan x) = \dfrac{1}{\sqrt{1+x^2}}\\ \sin(\arctan x) = \dfrac{x}{\sqrt{1+x^2}}$

To find cos a:

$\cos a\\ = \cos(\arcsin(\sin a))\\ =\sqrt{1-\sin^2 a}\\ =\sqrt{1-\left(-\dfrac{21}{29}\right)^2}\\ =\sqrt{\dfrac{29^2 - 21^2}{29^2}}\\ =\sqrt{\dfrac{20^2}{29^2}}\\ =\dfrac{20}{29}$

To find cos b:

$\cos b\\ =\cos(\arctan(\tan b))\\ =\dfrac{1}{\sqrt{1+\tan^2 b}}\\ =\dfrac{1}{\sqrt{1+\dfrac{576}{49}}}\\ =\dfrac{1}{\sqrt{\dfrac{625}{49}}}\\ =\dfrac{1}{\dfrac{25}{7}}\\ =\dfrac{7}{25}$

To find sin b:

$\sin b\\ =\sin(\arctan(\tan b))\\ =\dfrac{\tan b}{\sqrt{1+\tan^2 b}}\\ =\dfrac{-\frac{24}{7}}{\sqrt{1+\dfrac{576}{49}}}\\ =\dfrac{-\frac{24}{7}}{\sqrt{\dfrac{625}{49}}}\\ =\dfrac{-\frac{24}{7}}{\frac{25}{7}}\\ =-\dfrac{24}{25}$

cos a = 20/29, cos b = 7/25, sin b = -24/25, sin a = -21/29

To find cos(a + b):

$\cos(a+b)\\ =\cos a\cos b - \sin a \sin b\\ = \left(\dfrac{20}{29}\right)\left(\dfrac{7}{25}\right)-\left(-\dfrac{21}{29}\right)\left(-\dfrac{24}{25}\right)\\ = \left(\dfrac{140}{725}\right) - \left(\dfrac{504}{725}\right)\\ =-\dfrac{364}{725}$

Dan and Phil is YouTuber.......

I didn't know that there were such YouTubers in YouTube. Thanks for the information.

Never heard of them though.....

Btw what is ship????

$4^x = 4096\\ 2^{2x} = 2^{12}\\ 2x = 12\\ x = 6$

$oneplusone\\ = o^2n^2e^2plus$

Nope..... This is the correct answer......

Note that: $\boxed{\color{BurntOrange}{1^2 + 2^2 + 3^2 + 4^2 + ... + n^2=\dfrac{n(n+1)(2n+1)}{6}}}$

$\color{BurntOrange}{2^2 + 4^2 + 6^2 + ... + 20^2\\ =(1\times 2)^2 + (2\times 2)^2 + (3\times 2)^2 + ... + (10\times 2)^2\\ =1^2\times 2^2 + 2^2\times 2^2 + 3^2\times 2^2 + ... + 10^2 \times 2^2\\ = 2^2(1^2 + 2^2 + 3^2 + ... + 10^2)\\ =2^2\left(\dfrac{(10)(11)(21)}{6}\right)\\ =\left(\dfrac{1}{3}\right)(2)(10)(11)(21)\\ =(7)(2)(10)(11)\\ =(14)(10)(11)\\ =(154)(10)\\ =1540}$

$1\dfrac{5}{18}$.

PPAP.

DarkKnight:

Tonight? It is daytime for me!

This is an international forum.

LOL

~The smartest cookie in the world.

54, 64.8.

Because $34.5\div 30 = 18.4\div16=103.5\div90=1.15$

But $64.8\div 54 = 1.2$

~The smartest cookie in the world. :)