### Video Transcript

In the figure, the length of ๐ด๐ถ is 3.5. What is the length of ๐ด๐ต? Give your answer to two decimal places.

In this question, weโre given a figure of triangle ๐ด๐ต๐ถ, and weโre told that the length of side ๐ด๐ถ is 3.5. We can add this to our diagram. We need to determine the length of the side ๐ด๐ต. We need to give our answer to two decimal places. Thereโs a few different ways we could go about determining the length of ๐ด๐ต.

To do this, we note that in our diagram weโre given the point ๐ท which makes two right triangles. We have right triangle ๐ด๐ท๐ต and right triangle ๐ด๐ท๐ถ. And if we add the right angle for triangle ๐ด๐ท๐ถ, we note that we have one known side length for this triangle and we know one of the nonright angles of the triangle. This means we can determine the side lengths of this triangle by using right-triangle trigonometry. And in particular, we can use this to find the length of ๐ด๐ท. And then, weโll be able to use right-triangle trigonometry on our other right triangle ๐ด๐ท๐ถ to determine the length of ๐ด๐ต.

The first thing weโre going to need to do is label the sides of our right triangle ๐ด๐ท๐ถ. Letโs start with the hypotenuse. Itโs the longest side in the right triangle. Itโs the one opposite the right angle. In this case, thatโs side ๐ด๐ถ. Next, we should label the side we want to calculate; thatโs ๐ด๐ท. We can see in our diagram itโs opposite the angle of 41 degrees. So we can label this as the opposite side. Finally, although itโs not necessary, we can also see that side ๐ท๐ถ is adjacent to our angle of 41 degrees. So we can label this as the adjacent side.

Now that weโve labeled the sides of this right triangle, weโre ready to apply right-triangle trigonometry to determine the length of side ๐ด๐ท. And we can recall the following acronym: SOH CAH TOA. This will help us determine which trigonometric ratio we need to use. In our diagram, we want to determine the opposite side, and we know the length of the hypotenuse. So we need to use the trignometric ratio, which relates the opposite side to the hypotenuse. And this is the sine function. We can recall if ๐ is the angle in a right triangle, then the sin of ๐ will be equal to the length of the opposite side to angle ๐ divided by the length of the hypotenuse. Now, weโd substitute the values from our right triangle ๐ด๐ท๐ถ into this equation. We get sin of 41 degrees is equal to the length of ๐ด๐ท divided by 3.5.

Now, we can solve for ๐ด๐ท by multiplying both sides of our equation through by 3.5. We get that the length of side ๐ด๐ท is 3.5 times the sin of 41 degrees. And now we could use our calculator to evaluate this expression. However, remember, weโre trying to find the length of side ๐ด๐ต. And weโre going to need to use the length of side ๐ด๐ท to do this, so itโs easier to use the exact value. So weโll leave this expression as it is.

Letโs now sketch triangle ๐ด๐ต๐ท. We have the triangle ๐ด๐ท๐ต is a right triangle. We know one of the nonright angles of this triangle and we also know one of the side lengths. So we can determine the length of this triangle by using trigonometry. Once again, weโll start by labeling the sides of this triangle. Weโll start with the hypotenuse, the longest length of the triangle, the one opposite the right angle. In this case, thatโs ๐ด๐ต. Next, we see that the side ๐ด๐ท is the one opposite our angle of 63 degrees. So we can label this as the opposite side. And finally, although itโs not necessary, we can see that the side ๐ต๐ท is the one adjacent to the angle of 63 degrees. So we can label this as the adjacent side.

We now need to determine which trigometric ratio we need to use. And to do this, we will once again use our acronym of SOH CAH TOA. We have a very similar story. However, it is slightly different in this case. In triangle ๐ด๐ต๐ท, we know the length of the opposite side to our angle. However, we want to determine the length of the hypotenuse. So weโll once again be using the sine ratio. Now, all we need to do is substitute the values from triangle ๐ด๐ต๐ท into the sine ratio. We get sin of 63 degrees will be equal to 3.5 sin of 41 degrees divided by the length of ๐ด๐ต.

Now, all thatโs left to do is rearrange this equation for ๐ด๐ต. Weโll multiply both sides of our equation through by ๐ด๐ต and divide both sides of the equation through by sin of 63 degrees. This gives us that the length of ๐ด๐ต is equal to 3.5 times the sin of 41 degrees divided by the sin of 63 degrees. And now, we can evaluate this expression by using our calculator, where we need to make sure itโs set to degrees mode. If we do, we get 2.577 and this expansion continues. Well, itโs worth pointing out we could also give this the units of length units since we know this represents the length.

Finally, the question wants us to write our answer to two decimal places. To do this, we look at the third decimal digit of our number, which is seven. Since this is greater than or equal to five, we need to round this value up. And in doing so, we get our final answer. Given in the figure that ๐ด๐ถ is 3.5, we were able to conclude the length of ๐ด๐ต is 2.58 to two decimal places.