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# Trig challenge problem: verify identities

Video transcript

The diagram below contains
parallelogram ABCD, so that's parallelogram
ABCD, and triangle EFG. And they tell us
which of the angles have a measure of 90 degrees,
and they label it here, as well. And they tell us
which of the angles have a measure of 31 degrees,
and they label it here, as well. Which of the equalities
must be true? I encourage you to
now pause this video and try to figure
this out on your own. So now let's look at
this first statement, tangent of angle ADC. So let's think about the
tangent of angle ADC. So this is angle ADC
right here-- ADC. To remind ourselves the
definition of tangent, we'll break out sohcahtoa. Sine is opposite
over hypotenuse, cosine-- soh cah-- cosine
is adjacent over hypotenuse, toa-- tangent is
opposite over adjacent. So what is the opposite
side to this angle? So we're dealing with this
right triangle, triangle ADC. Let me highlight
it so that we know we're dealing with this right
triangle right over here. That's the only right triangle
that angle ADC is part of. And so what side is
opposite angle ADC? Well, it's side CA, or I
guess I say AC, side AC. So that is opposite. And what side is adjacent? Well, this side, CD. CD, or I guess I could call it
DC, whatever I want to call it. DC, or CD, is adjacent. Now how did I know that
this side is adjacent and not side DA? Because DA is the hypotenuse. They both, together, make up
the two sides of this angle. But the adjacent side
is one of the sides of the angle that is
not the hypotenuse. AD or DA in the
sohcahtoa context we would consider to
be the hypotenuse. For this angle, this is
opposite, this is adjacent, this is hypotenuse. Tangent of this angle is
opposite over adjacent-- AC over DC. Now is that what
they wrote here? No. They wrote AC over EF. Well, where's EF? EF is nowhere to be seen
either in this triangle, or even in this figure. EF is this thing
right over here. EF is this business
right over here. That's EF. It's in a completely
different triangle in a completely
different figure. We don't even know what
scale this is drawn at. There's no way the
tangent of this angle is related to this somewhat
arbitrary number that's over here. They haven't labelled it. This thing might be a million
miles long for all we know. This thing really
could be any number. So this isn't the case. We would have to relate it to
something within this triangle, or something that's
the same length. So if somehow we
could prove that EF is the same length as DC,
then we could go with that. But there's no way. This is a completely different
figure, a completely different diagram. This is a similar
triangle to this, but we don't know anything
about the lengths. A similar triangle just lets
us know that the angles are all the same, or that the ratio
between corresponding sides might be the same,
but it doesn't tell us what this number right
over here, doesn't tell us that this side is
somehow congruent to DC. So we can't go with this one. Now let's think about
the sine of CBA. So the sine-- let me do
this in a different color. So the sine of angle CBA. So that's this angle
right over here, CBA. Well, sine is opposite
over hypotenuse. I guess let me
make it clear which triangle we're looking at. I'll do this in yellow. We're now looking at this
triangle right over here. The opposite side is AC. That's what the
angle opens up into. So it's going to be equal to AC. And what is the hypotenuse? What is the hypotenuse here? Well, the hypotenuse--
so let me see, it's opposite over hypotenuse--
the hypotenuse is BC. It's the side opposite
the 90 degree side. So this, it's BC. Sine is opposite over
hypotenuse, so over BC. Is that what they
wrote over here? No. They have DC over BC. Now what is DC equal to? Well, DC is this. And DC is not-- there's no
evidence on this drawing right over here that DC is
somehow equivalent to AC. So given this information
right over here, we can't make this
statement, either. So neither of these are true. So let's make sure
we got this right. We can go back to
our actual exercise, and we get-- oh, that's
not the actual exercise. Let me minimize that. This is neither
of these are true. And we got it right.