Question Video: Finding the Length of a Side in a Right Triangle Using Trigonometric Identities | Nagwa Question Video: Finding the Length of a Side in a Right Triangle Using Trigonometric Identities | Nagwa

# Question Video: Finding the Length of a Side in a Right Triangle Using Trigonometric Identities Mathematics • Third Year of Preparatory School

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The circumference of a circle with a center ๐ is 259 and the line segment ๐๐ is tangent at ๐. Calculate the length of the line segment ๐๐ to the nearest hundredth.

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### Video Transcript

The circumference of a circle with a center ๐ is 259 and the line segment ๐๐ is tangent at ๐. Calculate the length of the line segment ๐๐ to the nearest hundredth.

Well, the first thing we do with this type of question is mark on any values we know. Well, the first thing weโre told is that the circumference of the circle is 259. Well, we know that thereโs a formula for the circumference of a circle. And this is that ๐ is equal to ๐๐. But how is this gonna be useful? Well, if we take a look at our sketch, we can see that ๐๐ is gonna be the diameter and thatโs because we know that the center is ๐. So, therefore, what we can do is use our formula and the information we know to work out the diameter. Because what we can say is that 259 is gonna be equal to ๐๐. Well, this is going to give us that 259 divided by ๐ is equal to ๐. So thatโs our diameter.

Well, what we could have done now is calculated what this is. But we leave this in this form for accuracy because it prevents any rounding issues later in the problem. Okay, so we have this side marked on, or weโve got ๐๐ โ our diameter โ marked on. But is there anything else we can do? Well, weโre told that the line segment ๐๐ is tangent at ๐. But what does this mean? Well, we know that if a line is tangent to a circle at a point, then at this point a right angle is gonna be formed between the tangent and the radius, which Iโve shown here. Once again, we know that ๐๐ is a radius because we know that ๐ is the center of the circle.

Okay, great. So, now, what do we need to do? Well, in the question, we can see that what weโre trying to do is find the length of the line segment ๐๐. Well, what we, in fact, have now is a right triangle, where we have one side, then a side we want to find, and then weโve got an angle. So, therefore, because we have these things and because weโve got this angle, what weโre gonna use is our trigonometric ratios to help us solve this problem and find the line segment ๐๐. Well, the first thing we do if weโre gonna use trigonometry is label our triangle.

So weโve got the hypotenuse, which is the longest side opposite the right angle. Then, weโve got the opposite, which is opposite the angle that weโve been given. And then weโve got the adjacent, which is between the right angle and the angle weโve been given. And also, itโs adjacent to the angle weโve been given. Well, whenever weโre solving a trigonometry problem, weโve got a stepped approach. Our first step was label, which weโve done. And the second step is to decide which one of our trigonometric ratios weโre going to use.

And to help us do that, what we have is a memory aid, which is SOHCAHTOA. And what this tells us is that sine of an angle is equal to the opposite divided by the hypotenuse. The cosine of an angle is equal to the adjacent divided by the hypotenuse. And the tangent of an angle is equal to the opposite divided by adjacent. Well, if we take a look at our problem, we can see that we know the opposite and we want to find the adjacent because thatโs the line segment ๐๐. So therefore, if we look at our SOHCAHTOA, we can see the ๐ and ๐ด, so the opposite and the adjacent, are part of the tangent ratio. So, therefore, weโre gonna use tan ๐ is equal to the opposite divided by the adjacent.

So now, step three or the next step is to substitute in the values weโve got. So when we do that, what weโre gonna get is that tan 30 is equal to 259 over ๐ over ๐ด, which is our adjacent. So now, what weโre gonna do to rearrange and make ๐ด the subject is multiply by ๐ด and then divide by tan 30. So now, what weโre gonna do is put it in the calculator because weโve got the ๐ด, which is our line segment ๐๐, is equal to 259 over ๐ divided by tan 30. So, therefore, we know that the adjacent ๐ด is gonna be equal to 142.79418 etc. But we havenโt finished here cause what weโre looking for is the answer to the nearest hundredth.

Well, if weโre looking to the nearest hundredth, itโs gonna be the second decimal place because the first decimal place is the tenth. So, weโve drawn a line here. And then, the number to the right of this is gonna be the deciding number. And because itโs less than five, then what weโre gonna do is round down. So, therefore, what we can say is that the length of the line segment ๐๐ is 142.79 and thatโs to two decimal places or the nearest hundredth.

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