WEBVTT
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[MUSIC PLAYING]
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Having looked at the inverse
of the tangent function,
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we're now going to look at the
other inverse trig functions.
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Let's begin with the case
of the sine function.
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As we know, that's what
its graph looks like.
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Now we need to
restrict the domain
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so that on that
restricted domain
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the sine function is
injective and also attains
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all its relevant values.
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The obvious choice
here, and it's
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the choice that is always
made, is the interval minus pi
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over 2 to pi over 2.
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We see that on that interval,
the function is strictly
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increasing.
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And it takes all the values
between minus 1 and plus 1.
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So we consider that
function with that domain
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and that interval.
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It's then a bijective function.
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Therefore, we can define
an inverse function.
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And the name we give to
that inverse function
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is arcsine, also sometimes
denoted inverse sine.
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So it's a function that
maps the other way.
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It maps the interval
minus 1, 1 back
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to the interval minus
pi over 2, pi over 2.
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To summarize then, we've
looked at the sine function
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between minus pi over 2 and
pi over 2, restricted domain.
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And we've constructed
the inverse function
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in the following way.
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We've said that for
each t in minus 1, 1,
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there is a unique x in that
interval for which t is equal
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sine x.
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There's the t,
and there's the x.
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And we're defining arcsine of t
to equal that value of x, also
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called inverse sine function
and denoted this way.
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What does the graph
of arcsine look like?
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Rather easy to find.
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You apply the usual--
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one of the two usual methods
we have geometrically
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to find the graph.
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You get this.
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As you can see,
arcsine is defined
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on the interval minus 1,
plus 1, and not otherwise.
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What about the
inverse of cosine.
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Well, it's going
to be very similar.
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One difference
though is that you
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can't take the interval
minus pi over 2 to pi
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over 2, because
on that interval,
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the cosine is not
injective, as you can see.
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So you shift the interval.
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You take the interval 0, pi,
instead for inverting cosine.
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On 0, pi, you see that cosine
is strictly decreasing,
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and hence injective.
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So you obtain an
inverse function.
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It's labelled arccosine.
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And it maps minus 1, 1
to the interval 0, pi.
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Easy enough to find what
the graph looks like.
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It turns out to be like this.
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Notice that arccosine,
also called inverse cosine,
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is defined just on the
interval minus 1 to plus 1.
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There are a few forgotten
functions that we're not
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going to discuss in any detail.
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As you know, our
discussion of trigonometry,
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which is now drawing
to a close, has
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defined three functions, sine,
cosine, and tangent, as well as
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their inverses.
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But depending on what
continent you live on,
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you may also run into other
trigonometric functions
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that are current in
trigonometry, for example,
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cotangent.
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Cotangent x means the cosine
over the sine, equivalently 1
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over the tangent.
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The secant of x is
1 over the cosine.
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And the cosecant
is 1 over the sine.
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And these functions
also have inverses
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that can be defined in similar
ways to what we've done.
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And if you're dealing with
all of these functions,
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then there are of course,
even more identities
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that you might have to know.
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For example, it's easy to prove
that tangent squared plus 1 is
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equal to secant squared.
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However, it is possible
to do everything
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you want in trigonometry
without dealing
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with these extra functions,
as some countries do.
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Another class of functions
we have not discussed
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is related to the exponential
function, e to the x.
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These are the so-called
hyperbolic functions.
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They were defined and
introduced in the 19th century.
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They're used a lot by
engineers, for example,
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because certain differential
equations involve
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these as solutions.
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What are these functions?
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Well, here they are.
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They're read usually
as hyperbolic sine,
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hyperbolic cosine, and
hyperbolic tangent.
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And as you see,
they're completely
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defined in terms of the
exponential function.
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And they also admit inverses,
which have some uses.
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And they have a whole bunch
of identities of their own.
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Actually, they're rather
amusing in the sense
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that the identities for
these hyperbolic functions
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look a lot like
trigonometric identities
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with occasionally a
sign being different.
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There's a circular
identity, for example,
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that says that
hyperbolic cosine squared
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minus hyperbolic sine
squared is equal to 1.
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Now, there are so many
functions and so little time
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that we're not going
to do them all.
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But I can assure you
that we have done
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the ones that count the most.
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