### Video Transcript

Find 17 tan of π΅ cos of πΆ over sin squared of πΆ plus cos squared of π΅, given that
π΄π΅πΆπ· is an isosceles trapezoid, where line segment π΄π· is parallel to line
segment π΅πΆ, π΄π· equals eight centimeters, π΄π΅ equals 17 centimeters, and π΅πΆ
equals 24 centimeters.

And then weβve been given a diagram of the isosceles trapezoid. So letβs first remind ourselves what we mean when we talk about an isosceles
trapezoid. Well, in an isosceles trapezoid, the two nonparallel sides are equal in length. This also means that the angles that these nonparallel sides make with the two
parallel sides are equal.

Now, in this question, weβre looking to find an expression for various trigonometric
functions of angles π΅ and πΆ. So letβs define angles π΅ and πΆ to both be equal to π. We know the trigonometric ratios for a right triangle with included angle π tell us
that sin of π is equal to opposite over hypotenuse. cos of π is equal to adjacent divided by the hypotenuse. And tan of π is equal to opposite divided by adjacent. So where are we going to find a right triangle in our diagram? Well, we can add a perpendicular directly up from point π΄. We know that this triangle has an included angle of π, and it has a hypotenuse of 17
centimeters. So letβs work out the length of the side adjacent to the angle π, and weβll call
that π₯ centimeters.

Since the trapezoid is isosceles, we know if we add a second perpendicular directly
up from point π·, we end up with another triangle with an adjacent side of π₯
centimeters. Then, we also have a rectangle, so the remaining portion of line segment π΅πΆ must be
eight centimeters in length. We can therefore say that since line segment π΅πΆ consists of three portions, π₯,
eight, and π₯, which sum to make 24 centimeters, two π₯ plus eight must be equal to
24. We now have an equation we can solve for π₯. We begin by subtracting eight from both sides, so two π₯ equals 16. Then, dividing through by two, we find π₯ is equal to eight.

And so we now have a right triangle for which we know two of the dimensions. We are going to work out the third dimension since at some point weβre going to
calculate sin of π, cos of π, and tan of π. Here is the triangle weβre interested in. We can use the Pythagorean theorem to work out the missing length in this triangle,
the side opposite the included angle. We might alternatively notice that we have a Pythagorean triple, but letβs use the
Pythagorean theorem to check.

Letβs call the length of the side that weβre trying to find π¦ centimeters. Then, by the Pythagorean theorem, the sum of the squares of the two shorter sides
must be equal to the square of the hypotenuse. So eight squared plus π¦ squared is equal to 17 squared. 64 then plus π¦ squared equals 289. By subtracting 64 from both sides, we find that π¦ squared is equal to 225. Then, taking the positive square root of 225, we find that π¦ is equal to 15. And so we now have the third side of the triangle. And there is a Pythagorean triple we might recognize. Eight squared plus 15 squared is 17 squared.

We now have everything we need to calculate the value of tan of π΅, cos of πΆ, sin of
πΆ, and cos of π΅. tan of π΅ is of course tan π, and we know that tan π is opposite over adjacent. In our triangle, the opposite side is 15 centimeters and the adjacent is eight. So tan of π is 15 over eight. Next, we need to find the value of cos of πΆ and cos of π΅. But since both πΆ and π΅ are π, we simply need to find the value of cos of π. Now, of course, thatβs adjacent divided by hypotenuse. So itβs eight over 17. cos of π is eight seventeenths. Finally, we need to know the value of sin of πΆ. So thatβs sin of π, where sin of π is opposite over hypotenuse. So sin of π is 15 over 17.

Letβs now find the value of 17 tan of π΅ cos of πΆ over sin squared of πΆ plus cos
squared of π΅. The numerator is 17 times tan of π΅, 15 over eight, times cos of πΆ, eight over
17. Then, the denominator, sin squared of πΆ plus cos squared of π΅, is eight
seventeenths squared plus 15 over 17 squared.

Now, before we try and evaluate this in our head, we might notice that we can
simplify somewhat. First, on our numerator, we can simplify by dividing through by 17. Similarly, we can divide through by eight, and that leaves simply 15 on the numerator
of our expression. Next, remember that eight squared plus 15 squared is 17 squared. Thatβs our Pythagorean triple. So the denominator of this expression is actually going to equal one. This means we get 15 over one, which of course is simply equal to 15.

So the value of 17 tan of π΅ cos of πΆ over sin squared of πΆ plus cos squared of π΅
is 15.