Question Video: Using Trigonometric Ratios to Find the Length of the Side Opposite the Angle Mathematics

Find the length of the line segment ๐ต๐ถ, giving the answer to two decimal places.


Video Transcript

Find the length of the line segment ๐ต๐ถ, giving the answer to two decimal places.

In this question, weโ€™re asked to determine the length of the line segment ๐ต๐ถ. And we can see in the diagram that ๐ต๐ถ is the side length in a right triangle. We need to give our answer to two decimal places. To answer this question, letโ€™s start by saying that the line segment ๐ต๐ถ has a length of ๐‘ฅ. We can now see weโ€™re trying to determine the length of the side in a right triangle where we know one of the non-right angles of the right triangle and one of the other side lengths. This means we can determine the value of ๐‘ฅ by using right triangle trigonometry.

And we recall the first thing we should do when trying to use right triangle trigonometry is label the side lengths of the triangle. First, we should label the hypotenuse. Itโ€™s the longest length in the right triangle, which is the one opposite the right angle. And in this case, we can see thatโ€™s the side ๐ด๐ถ. This is called the hypotenuse of the right triangle.

Now we need to label the other two sides of our right triangle. And we do this by considering their position in relation to our known angle. In particular, we can see that the side ๐ต๐ถ is opposite our angle of 47 degrees. So we call the side ๐ต๐ถ the opposite side. Finally, although itโ€™s not necessary in this question, we can label the side ๐ต๐ด. Since the side ๐ต๐ด is next to the angle of 47 degrees and not the hypotenuse, we call it the adjacent side.

And now that weโ€™ve labeled all three sides of the right triangle relative to their position of the angle of 47 degrees, we can start applying right triangle trigonometry. To do this, weโ€™ll start by recalling the following acronym: SOH CAH TOA. This acronym can help us determine which of the three trigonometric ratios we need to use to answer the question. And we can do this by considering the values we know or want to find. First, we know the hypotenuse of this right triangle. And second, we want to determine the length of the opposite side to our angle. In other words, we want to find the trigonometric ratio linking the opposite side to the hypotenuse.

And we can see that this is the sine function. If ๐œƒ is the angle in a right triangle, then the sin of ๐œƒ is the ratio of the length of the opposite side to angle ๐œƒ divided by the length of the hypotenuse. Now, all we need to do is substitute the values in our right triangle into this equation. Our value of ๐œƒ is 47 degrees. The length of the hypotenuse is 15 centimeters. And the length of the opposite side is ๐‘ฅ. Therefore, the sin of 47 degrees is equal to ๐‘ฅ divided by 15.

Now, all we need to do is solve this equation for ๐‘ฅ. Weโ€™ll do this by multiplying both sides of the equation through by 15. This gives us that ๐‘ฅ is equal to 15 times the sin of 47 degrees. And we can evaluate this by using a calculator. To do this, we type the expression into our calculator, and we need to remember to set our calculator to degrees mode. This would give us that ๐‘ฅ is 10.970, and this expansion continues. And itโ€™s important to remember in the diagram weโ€™re given that the lengths were in centimeters, so ๐‘ฅ is measured in centimeters.

Finally, the question wants us to give our answer to two decimal places. So weโ€™ll look at the third decimal place of ๐‘ฅ, which is zero. Since this is less than five, we need to round this value down, giving us 10.97 centimeters, which is our final answer. Therefore, we were able to determine the length of the line segment ๐ต๐ถ to two decimal places. It was 10.97 centimeters.

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