i have a conundrum this exponential equation has no real solutions but i can solve it
however this exponential equation has real solutions but i cannot solve it so let's
solve this exponential equation first and you know it we will have to use the lambert
w function but before that we will have to take the natural log on both sides because this way
we can put the power to the front and let me write down the sine x in red and we have this ln of sine
x and don't worry we have to go to a complex world anyway so don't worry about having negative inside
of the natural log and the right hand side is just ln2 now we are going to write the sine x as
e to the ln of sine x power because we have this right here already and as you can see this and
that match and we have the e right here so we can use the lambda w function very nice and this is
still l one two okay let's go ahead and just take the lambda w function on both sides right here and
also right here and of course on the left-hand side we will just get this back so we get ln of
sine x being equal to the lambda w function of ln2 and then i would like to get rid of the ln
with just 2 e to the power e to the power and now as we can see we must have sine x being equal to
e to the lambda w function of ln2 and here is the reason why that we have to go to a complex world
not just because it's fun but the reason is because if you enter this number from alpha
you will get approximately 1.5596 which is bigger than 1 which is outside of the
range of sine x that's okay we will get complex solution for this so let's continue we are going
to use the complex definition for sine x in terms of e to the ix minus e to the negative ix
over 2 i and then let's write everything in the example so we'll put this down and you can see
how crazy the answer looks like later on okay this right here is going to be e to the w of ln2
next let's multiply everybody by 2i and also e to the ix because that way we can get rid of that
okay take this multiplied by that the two i can so so take this multiplied by that we will have
e to the ix square and then take this multiplied by that it's just minus one but let me put it down
right here and when we do this times that well it's on the right hand side let's put that to
the left-hand side so we'll get minus and then we have two i and also this which is e to the w
of l and 2 and then we have the e to the ix and of course now this is just going to be equal to zero
have a look what's this this square and then we have this and then let's know this
so it's a quadratic equation in this so we can use the quadratic formula so we are going to have e to
the ix being equal to negative b which is this so let's put down negative 2 i e to the w of ln2
and then plus or minus and open the square root we have to do b square which is that
again so let's put down negative 2i e to the w of ln2 squared and then minus 4 times a and c
a is 1 and c is negative 1. so let's go ahead and put them down right here
and then all divided by 2 times a which is 2 times 1. perfect on the left hand side
we still have e to the ix and on the right-hand side of course let's simplify this a little bit
this is positive 2i e to the w of ln2 plus row minus and here check this out negative square is
positive but we have a i squared which is going to give us negative so in fact the coefficient
here is negative 4 and here we also have a negative 4. so let's factor out negative 4
but that's inside the square root so in fact that will be a 2i right here and then of course we have
this left but we have to do that to the second power so it becomes e and of course we multiply
the power so we have the two and w of ln2 and of course we took out the negative 4 so we have the
negative 1 right here and all divided by 2. and now cancel cancel cancel and of course we can also
factor out the i and in the meantime let's also take the natural log so this right here becomes
ix being equal to again we took the natural log right so we have the natural log here
and then we have the i being factored it times just this plus minus that which is e
okay what's next you can divide both sides by i but a better way is multiply all this equation by
negative i because that will give us negative one multiplied by negative
negative that's going to give us positive x so ladies and gentlemen x is going to be negative
i so i'll put this down in blue negative i but the thing is that you know you see
here we have ln of this times that so we're going to break them apart so it becomes negative i times
here we will have this is lni we are going to take care of that in a second and then
we are going to add ln of the rest which is going to be just all that so
and then close that close that i did close the parenthesis okay
okay so what is l and i well if you look at the complex plane i is right here the angle
is pi over two and then um radius the distance is 1. so this right here is the same as e to the
pi over 2 i this will give us i if you put this right here e and l will cancel
so in fact this right here is just pi over 2 so pi over 2 and of course we have the i
okay all right and then of course just finally fix everything x is going to p
i times i is negative one but i have another negative so it's positive so we have positive
pi over two and then do this times that so we will have a minus i and then all that stuff again
close parentheses yep but that's not it yet this is not it yet even though you can produce a lot
of solutions based on the branch of w but you know x came from sine right here right
so we will have to do what the period of sine is 2 pi which is how to keep adding
it so i would just say we have all this stuff and we have to add 2 n pi where n is an integer
but anyway this right here
is indeed the solution to that equation i told you solvable in the complex world which is pretty
crazy and yeah this is like the super version of sine x is equal to 2 right so now let's take a
look at the equation sine x to the cosine x power it's equal to 2. and you might be wondering how is
this possible to have real solutions because we know the absolute value of x in the real world
it's always going to be less than or equal to 1. so is for cosine x and now see we are saying that
the absolute value of this is at most one raised to the absolute value of this is at most one and
somehow we can legitimately and that was two how's that possible well don't worry let me explain this
is what i was thinking when i was coming with this equation let me show you guys the unit circle it
helps keep in mind everything is in the real world now with that said we need to have the base to be
positive and for sin x to be positive that means we have to be in the first and also the second
quadrant so now keep in mind the base is positive and it's going to be less than or equal to one
so let me just put this down we want to have zero less than sine x and then less than one
how is this possible that we take this raised to a power and we end up with 2.
well it is possible if the power here is negative and in fact cosine x will be negative if we are
in the second quadrant so perfect and in fact if you are like go ahead and just
graph the equation sine x to the cosine x power you will see the graph looks something like this
so this is y equals sine x to the cosine x power and of course uh it crosses two right
here and it has infinitely many of these curves but i'm not going to graph them all but now
i will have to tell you guys the sinus i don't know how to find the real solution for this i
tried i really did let me show you right here sine x for the base we know cosine can be written as
plus or minus square root of 1 minus sine square x and we say that's equal to 2.
and i tried what we did earlier i took the natural log on both sides so i can bring the
the dusting in the front and all that stuff but it's getting too crazy you know
this is just like an unfortunate that this equation is actually so nice especially you
have real solutions but i don't know how to solve it if you guys know please let me know