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anonymous
 one year ago
please help... FAN AND MEDAL
anonymous
 one year ago
please help... FAN AND MEDAL

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freckles
 one year ago
Best ResponseYou've already chosen the best response.3which one do you looking at?

freckles
 one year ago
Best ResponseYou've already chosen the best response.3ok well lets look at: \[\cos(2x)=\frac{1}{\sqrt{2}} \\ \text{ or also known as } \\ \cos(2x)=\frac{\sqrt{2}}{2}\] Before we solve this one, do you know how to solve: \[\cos(\theta)=\frac{\sqrt{2}}{2}\] also are we solving in a specific interval ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the interval is between 0 and 360

freckles
 one year ago
Best ResponseYou've already chosen the best response.3ok if we have \[0<x<360 \\ \text{ and we \let } theta=2x \\ \text{ then } x=\frac{\theta}{2} \\ \text{ and so we have } \\ 0<\frac{\theta}{2}<360 \\ \text{ multiplying both sides by 2 we have } \\ 0< \theta<720 \\ \text{ so you said } \cos(\theta)=\frac{\sqrt{2}}{2} \text{ has solutions } \\ \theta=45^o,315^o, 45^o+360^o,315+360^o \\ \text{ after simplifying } \\ \theta=45^o,315^o,405^o,675\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3now recall \[\theta=2x \\ \text{ so we have } \\ 2x=45^o ,315^o,405^o,675^o\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3just divide both sides by 2

freckles
 one year ago
Best ResponseYou've already chosen the best response.3ok and two more solutions

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[x=\frac{45^o}{2},\frac{315}{2}^o,\frac{405^o}{2},\frac{675^o}{2}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3we had to solve cos(theta)=sqrt(2)/2 in (0,720) since x was between 0 and 360

freckles
 one year ago
Best ResponseYou've already chosen the best response.3theta was 2 times more than x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ahh gotcha.. and thats the answer?

freckles
 one year ago
Best ResponseYou've already chosen the best response.3yes those 4 solutions right there at mentioned are the answers

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is that considered double identity angles?

freckles
 one year ago
Best ResponseYou've already chosen the best response.3we didn't use the double angle identity but we could have... \[\cos(2x)=\cos^2(x)(1\cos^2(x)) \\ \cos(2x)=2\cos^2(x)1 \\ \text{ so we have } 2\cos^2(x)1=\frac{\sqrt{2}}{2} \\ 2 \cos^2(x)=\frac{\sqrt{2}}{2}+1 \\ 2\cos^2(x)=\frac{2+\sqrt{2}}{2} \\ \cos^2(x)=\frac{2+\sqrt{2}}{4} \] but yucky this looks a whole bunch uglier

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ooo yeah definitely better the way you showed me.. thank you.. next one?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0btw i like how you actually explain whats going on its greatly appreciated

freckles
 one year ago
Best ResponseYou've already chosen the best response.3take the square root of both sides you will end solving two equations

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[\sin^2(x)=\frac{1}{2} \\ \text{ implies you have } \\ \sin(x)=\frac{1}{\sqrt{2}} \text{ or } \sin(x)=\frac{1}{\sqrt{2}}\] find the solutions to both equations and the solution will be the union of the sets of solutions you found

freckles
 one year ago
Best ResponseYou've already chosen the best response.3earlier you said you wanted the solutions between 0 and 360 \[\sin(x)=\frac{\sqrt{2}}{2} \text{ has solutions } x=45^o \text{ or } x=? \\ \sin(x)=\frac{\sqrt{2}}{2} \text{ has solutions } x=45^o+360^o \text{ or } x=?\] you still have 2 more solutions (1 for each equation)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3well you already have the solution

freckles
 one year ago
Best ResponseYou've already chosen the best response.3that was the 45+360 one

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i found 45, 315, 135 225

freckles
 one year ago
Best ResponseYou've already chosen the best response.3number 59 is very similar to 57

freckles
 one year ago
Best ResponseYou've already chosen the best response.3number 60 is a quadratic try solve 2u^21u=0 ( notice I just replaced cos(X) with u)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3solve for u you can use the quadratic formula if you want

freckles
 one year ago
Best ResponseYou've already chosen the best response.3then replace u with cos(x) and then solve for x

freckles
 one year ago
Best ResponseYou've already chosen the best response.3i will be back in like 30 minutes (sorry)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OMGGGG I GET IT! THANK YOU SO MUCH!!!!!!!! (hope the pie tastes bomb!)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3l messed up on the crust but the filling should taste great.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0haha aww its okay.. tbh ive never had pecan pie.. but i have one last question

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0just curious how old are you bc youre INCREDIBLE at math

freckles
 one year ago
Best ResponseYou've already chosen the best response.3There is some math superior to me (a lot of math actually) but I'm 30.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ahh gotcha... that makes me feel ALOT better haha.. im only a senior in high school and if you were my age and that mathematically inclined id be awfully embarrasssed lol

freckles
 one year ago
Best ResponseYou've already chosen the best response.3I would write purely in terms of sin(x) which will result in a quadratic equation again: \[\cos(2x)=\cos^2(x)\sin^2(x) \\ \cos(2x)=(1\sin^2(x))\sin^2(x) \\ \cos(2x)=12 \sin^2(x)\] you would be embarrassed?

freckles
 one year ago
Best ResponseYou've already chosen the best response.3oops I mean in terms of cos(x)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes i would bc compared to you im math illiterate...

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[\cos(2x)=\cos^2(x)\sin^2(x) \\ \cos(2x)=\cos^2(x)(1\cos^2(x)) \\ \cos(2x)=2\cos^2(x)1 \\ \text{ and } \sin^2(x)=1\cos^2(x)\] I forgot about that one term being cos(x) :p

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[\sin^2(x)+\cos(2x)\cos(x)=0 \\ \text{ is equivalent to } \\ 1\cos^2(x)+2\cos^2(x)1\cos(x)=0 \\ \text{ I replaced } \sin^2(x) \text{ with } 1\cos^2(x) \\ \text{ and I replaced } \cos(2x) \text{ with } 2\cos^2(x)1\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[1\cos^2(x)+2\cos^2(x)1\cos(x)=0 \\ \text{ combine like terms on the \left } \\ \cos^2(x)\cos(x)+11=0 \\ \cos^2(x)\cos(x)=0\] and actually this is tons easier to solve than using the quadratic formula this one is easily factorable on the the left hend side

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[a^2a=a(a1)\] \[\cos^2(x)\cos(x)=\cos(x)(\cos(x)1)\] you will have two equations to solve \[\cos(x)(\cos(x)1)=0 \\ \text{ gives you } \cos(x)=0 \text{ or } \cos(x)1=0\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3cos(x)=0 when x=90 or x=270 good job there the other equation can be written as cos(x)=1

freckles
 one year ago
Best ResponseYou've already chosen the best response.3one question is the interval in which we solve the equation (0,360) or [0,360) or (0,360] or [0,360]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so it would be 0 not 360

freckles
 one year ago
Best ResponseYou've already chosen the best response.3ok so [0,360) means we don't look at 360 just everything up to it and we do include 0 and things after you know until we get to 360 (which we do not include)

freckles
 one year ago
Best ResponseYou've already chosen the best response.3so instead of saying the solution to cos(x)=1 is 360 we will say 0

freckles
 one year ago
Best ResponseYou've already chosen the best response.3however if we were solving cos(x)=1 on [0,360] then we would say x=0 or x=360 since we can include both endpoints

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yesss gotcha! seriously thank you soooo much :)
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