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Trig challenge problem: verify identities

Sal is given a diagram with multiple right triangles and is asked to verify identities relating to trig ratios in those triangles. Created by Sal Khan.

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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.