Question Video: Using the Cosine Rule to Calculate an Unknown Length Involving Angles of Depression | Nagwa Question Video: Using the Cosine Rule to Calculate an Unknown Length Involving Angles of Depression | Nagwa

Question Video: Using the Cosine Rule to Calculate an Unknown Length Involving Angles of Depression Mathematics • Second Year of Secondary School

𝐸 and 𝐴 are two hot-air balloons flying at heights of 117 m and 84 m, respectively. The angles of depression at a point 𝐶 on the ground from 𝐸 and 𝐴 are 40° and 29°, respectively. Find the distance between the balloons giving the answer to the nearest meter.

05:19

Video Transcript

𝐸 and 𝐴 are two hot-air balloons flying at heights of 117 meters and 84 meters, respectively. The angles of depression at a point 𝐶 on the ground from 𝐸 and 𝐴 are 40 degrees and 29 degrees, respectively. Find the distance between the balloons giving the answer to the nearest meter.

Let’s start by identifying what we have and what we need. We have two right triangles with a side and an angle each, which means we can calculate the additional sides if necessary. We also have a nonright triangle, triangle 𝐴𝐶𝐸. And the distance between 𝐴 and 𝐸 is what we’re trying to find. We’re trying to find the length of one of the sides of a nonright triangle.

When working with nonright triangles, we know that we’ll be using either the sine rule or the cosine rule to find the value of the side labeled 𝑥. In order to do this, however, we need to find some additional information. First, we see that the line 𝐷𝐵 representing the ground is a straight line. And we know that angles in a straight line add to 180 degrees. If we subtract 40 and 29 from 180, we find the value of the angle that we’ve labeled 𝜃. It’s 111 degrees.

There’s still quite a bit of information we need to solve for the length of 𝑥. Let’s sketch separately the right triangle from our diagram to help us see what we can do next. It’s a right triangle with one angle and one side. This means we can use trigonometry to find the additional sides. The hypotenuse of this right triangle shares a side with the larger triangle 𝐸𝐶𝐴, which we’re trying to find a missing value for. To find the hypotenuse, we can use the sine relationship, where sin of 𝜃 is equal to the opposite over the hypotenuse. This means for the smaller right triangle, sin of 40 degrees will be equal to 117 over 𝐸𝐶.

Multiplying both sides of the relationship by 𝐸𝐶 gives us 𝐸𝐶 times sin of 40 degrees equals 117. And then dividing through by sin of 40 degrees gives us 𝐸𝐶 equals 117 over sin of 40 degrees. 𝐸𝐶 is equal to 182.01 continuing. You’ll notice here that we’re not going to round. Instead, we’ll use this exact value and save our rounding for the final step. This reduces the chance of error due to early rounding.

Let’s repeat this process with the smaller right triangle 𝐴𝐵𝐶, where the hypotenuse side 𝐴𝐶 is shared with the triangle 𝐴𝐶𝐸. Again, we know the opposite side length that we’re looking for the hypotenuse, which means we’ll use the sine relationship. sin of 29 degrees equals 84 over 𝐴𝐶. Multiplying both sides by 𝐴𝐶, 𝐴𝐶 times sin of 29 degrees equals 84. Dividing through by sin of 29 degrees, we find that 𝐴𝐶 equals 84 over sin of 29 degrees. 𝐴𝐶 is equal to 173.26 continuing.

Looking back at our diagram, we now have a nonright triangle in which we know two side lengths and a shared angle. Under these conditions, we can solve for that third side. This means to find the value of 𝑥, we can use the law of cosines, which tells us 𝑐 squared equals 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝑐, where 𝑐 is the side length we’re trying to find. And we know the other two sides and the angle opposite the side you’re trying to find.

Into this formula we can substitute 𝑒 in place of 𝑏 because we’re working with triangle 𝐴𝐸𝐶 instead of triangle 𝐴𝐵𝐶, where lowercase 𝑒 is the side opposite angle 𝐸, lowercase 𝑎 the side opposite angle 𝐴, and lowercase 𝑐 the side opposite angle 𝐶. Substituting these values in gives us 𝑥 squared equals 182.01 continuing squared plus 173.26 continuing squared minus two times 182.01 continuing times 173.26 continuing times cos of 111 degrees.

Remember when you’re entering these into your calculator to use the saved values of 𝐴 and 𝐸. This maintains the accuracy of our calculations. 𝑥 squared is then equal 85,620.14 continuing. Taking the square root of both sides will give us 𝑥 is equal to 292.6 continuing. Remember, we can disregard the negative square root here, as we’re dealing with distance. And 292.6 rounded to the nearest meter is 293. In the context of our question, this means that the hot-air balloons 𝐴 and 𝐸 are 293 meters apart.

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