Video Transcript
In this video, we will learn how to
find the value of a missing side length in a right triangle by choosing the
appropriate trigonometric ratio for a given angle. So what are these trigonometric
ratios?
When we have a right triangle, we
use the acronym SOH CAH TOA to help us remember the definitions of the trigonometric
ratios sine, cosine, and tangent. We say that the sin of 𝜃 is equal
to the opposite over the hypotenuse. The cos of angle 𝜃 is equal to the
adjacent side length over the hypotenuse. And the tan of angle 𝜃 is equal to
the opposite side length over the adjacent side length.
But in order to get these ratios
correct, we need to label the triangle correctly. And that means we’ll always
consider which angle we’re using. Here, this is our angle 𝜃. The opposite side is the side
directly opposite the angle concerned. The adjacent side is the side
between the angle and the right angle. And the hypotenuse is always the
longest side of the right triangle. It’s directly opposite the right
angle.
When we’re able to remember the
trigonometric ratios and correctly label our right triangle, we’re ready to start
looking at how to calculate unknown lengths of the right triangle. Here’s an example where we need to
find a missing side length.
Find 𝑥 in the given figure. Give your answer to two decimal
places.
The first thing we notice is that
this is a right triangle. We know an angle and a side
length. And that means to solve the
problem, we’ll need to use the trigonometric ratios. We remember the acronym SOH CAH
TOA. sin of 𝜃 equals the opposite over
the hypotenuse, cos of 𝜃 equals the adjacent over the hypotenuse, and tan of 𝜃
equals the opposite over the adjacent. Our starting point is always the
angle concerned. We’re given the angle 68
degrees. And so we can label the opposite
side side 𝑥; the adjacent side, the side that’s 11; and the hypotenuse is always
the side opposite the right angle.
Once we’ve labeled these sides, we
see that we’re given the length of the adjacent side. And we’re interested in the length
of the opposite side. Since we’re dealing with the
opposite and the adjacent, we’ll be looking at the tangent ratio. Since the tan of 𝜃 equals the
opposite over the adjacent, we plug in 68 degrees for the angle. The opposite side is the side we’re
trying to find, 𝑥. And the adjacent side measures
11.
In order to solve for 𝑥, we need
to isolate it to get it by itself. We can do that by multiplying both
sides of this equation by 11. 11 times tan of 68 degrees will
equal the side length 𝑥. From here, to solve, we’ll need to
use a calculator. We’ll enter 11 times tan of 68
degrees, and it will give us 27.22595 continuing. If your calculator does not return
this value, you should check and make sure that it’s set to degree mode and not to
radians.
The missing side of 𝑥 is
27.22595. We want it correct to two decimal
places. To round to the second decimal
place, we look to the right. Since there’s a five in the third
decimal place, we need to round up. And we’ll get that 𝑥 equals
27.23. As we aren’t given any units, it’s
fine to leave it in this format. 𝑥 equals 27.23.
Here’s another example. This time, we’re missing two of the
side lengths. And we need to solve for both of
the missing sides.
Find the values of 𝑥 and 𝑦,
giving the answer to three decimal places.
We notice that this is a right
triangle. We’re given an angle and a side
length, which means we can use trigonometric ratios to solve for the two missing
sides. Remembering the acronym SOH CAH
TOA, sin of 𝜃 equals the opposite over the hypotenuse, cos of 𝜃 equals the
adjacent over the hypotenuse, and tan of 𝜃 equals the opposite over the
adjacent. The key here is for us to label
this triangle correctly. And to do that, we’ll use the given
angle as our starting point.
We label the side lengths relative
to our given angle. 𝑦 is the side opposite to the
40-degree angle. 𝑥 is the adjacent side to the
40-degree angle. And the hypotenuse is always the
side opposite the right angle.
First, let’s try to solve for
𝑦. If we’re solving for 𝑦 and we know
the hypotenuse, we’ll use the sine ratio because the sin of 𝜃 is the opposite over
the hypotenuse. And that means we can say that the
sin of 40 degrees is equal to 𝑦 over 14. Since our goal is to solve for 𝑦,
we’ll multiply both sides by 14. And then we’ll see that 14 times
sin of 40 degrees equals 𝑦. When we plug that into our
calculator, we get 8.99902 continuing. If you don’t get this answer on
your calculator, then you should check and make sure that you’re operating in
degrees and not in radians.
We want our answer to three decimal
places. So we look to the fourth decimal
place, where there is a zero. That means we’ll round down. 𝑦 is equal to 8.999. And the units that we’re measuring
are centimeters. So we say that 𝑦 equals 8.999
centimeters.
Next, we need to solve for 𝑥. And we can solve for 𝑥 with two
different ratios. We could use the adjacent side and
the hypotenuse, which would be the cosine ratio. Or we could take what we found for
𝑦 and use that as the opposite side. And that would mean we would use
the tangent ratio because we would have the opposite and adjacent sides. In this case, let’s use the
hypotenuse as it will save us a little bit of writing.
We’re dealing with the cosine
ratio. We have cos of 40 degrees is equal
to 𝑥 over 14. We’ll multiply both sides by
14. 14 times cos of 40 degrees will
equal 𝑥. So 𝑥 will be equal to 10.72462
continuing. Rounded to the third decimal place
means we need to round up to 10.725. Again, the units here will be
measured in centimeters. And so we can say that 𝑥 is equal
to 10.725 centimeters and 𝑦 is equal to 8.999 centimeters, each to three decimal
places.
Notice how in these problems we’ve
been dealing with missing side lengths as the numerator of the fraction in the
ratio. Let’s look at an example where we
have a side length that ends up in the denominator of this ratio.
Find the values of 𝑥 and 𝑦,
giving the answer to three decimal places.
We have a right triangle. We’re given an angle and a side
length and asked to find the two missing sides. To do this, we’ll need our
trigonometric ratios. And to remember those, we’ll use
SOH CAH TOA. The sin of 𝜃 equals the opposite
over the hypotenuse. The cos of 𝜃 equals the adjacent
over the hypotenuse. And the tan of 𝜃 equals the
opposite over the adjacent. The key to solving these problems
consistently is to correctly label the triangle. And we label them relative to the
angle that we’re given. This is our starting point. The opposite side length is the
side length directly opposite this angle. The adjacent side is between this
angle and the right angle. And the hypotenuse is always
opposite the right angle.
Once a triangle is labeled, we’re
ready to identify which of the ratios we need. If we start by finding side length
𝑦, the hypotenuse, and we already know the opposite side, 28 centimeters, we need
to use the sine ratio, as the sine ratio involves the opposite side length and the
hypotenuse. The ratio would look like this. sin of 47 degrees is equal to 28
over 𝑦. When our variable is in the
denominator, it will take two steps to find the value.
The first thing we would do is
multiply both sides of the equation by 𝑦. When we do that, we get 𝑦 times
sin of 47 degrees equals 28. If the goal is to isolate 𝑦, then
at this point, we need to divide both sides of the equation by sin of 47
degrees. And then on the left we’ll just
have 𝑦, and on the right we’ll have 28 over sin of 47 degrees.
When we plug that into the
calculator, we get 38.28516 continuing. We need to round it to three
decimal places. This value rounds down to
38.285. The sides are being measured in
centimeters, so the units here would be centimeters. And that means we found one of the
missing sides.
To find the side length 𝑥, we’ll
have two choices. We could use the hypotenuse we just
found, 38.285. If we did that, we’d be dealing
with the adjacent side and the hypotenuse, which would be the cosine
relationship. Or we could use the 28-centimeter
side. In that case, we would be using the
opposite side and the adjacent side and would need the tangent ratio.
In this case, let’s practice having
the 𝑥-variable in the denominator. tan of 47 degrees equals 28 over
𝑥. To solve for 𝑥, we first multiply
both sides of the equation by 𝑥. Then, we can say that 𝑥 times tan
of 47 degrees equals 28. To isolate 𝑥, we divide both sides
of the equation by tan of 47 degrees. And so we say that 𝑥 equals 28
over the tan of 47 degrees, which gives us 26.11042 continuing. We round to the third decimal
place, and we get that 𝑥 is equal to 26.110. This is measured in
centimeters. And so we found the two missing
side lengths. To three decimal places, 𝑥 is
equal to 26.110 centimeters and 𝑦 is equal to 38.285 centimeters.
Let’s look at one final example
where we’re not given a diagram.
Find the length of the line segment
𝐴𝐶, given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where sin of 𝐶 equals nine over
16 and 𝐴𝐵 equals 18 centimeters.
In this case, the first step should
be to sketch a right triangle that meets these conditions. We have a right triangle. The right angle is at 𝐵, so we
label the right angle 𝐵. And then we add on 𝐴 and 𝐶. We’re told that 𝐴𝐵 measures 18
centimeters. And then we have this other piece
of information that sin of 𝐶 equals nine over 16. This tells us that the angle we’re
working with is angle 𝐶. And if we think of our acronym SOH
CAH TOA, we know that the sine of an angle is equal to the opposite over the
hypotenuse.
If the angle we’re considering is
𝐶, the opposite will be the side 𝐴𝐵, and the hypotenuse is always the side
opposite the right angle. It’s the side 𝐴𝐶. And so that ratio is nine over
16. The key thing to remember here is
that these relationships are ratios. And so sin of angle 𝐶 tells us
that for every nine units on the opposite side length, there will be 16 units on the
hypotenuse side length.
So we can say that if there are 18
centimeters on the opposite side, we know that nine times two equals 18. And when dealing with ratios or
fractions, if we multiply by two in the numerator, we need to multiply by two in the
denominator. 16 times two is 32. And so we can say that if the
opposite is 18, the hypotenuse must be 32. Line segment 𝐴𝐶 is the
hypotenuse, and it measures 32 centimeters.
Let’s now summarize the key points
from this video. When we have right triangles and we
need to solve for one of the sides, we have to remember the three trigonometric
ratios and then follow these steps. One, label the sides in the
triangle as the opposite, adjacent, and the hypotenuse, relative to the known
angle. Two, choose the correct
trigonometric ratio which links the known side to the unknown side using the acronym
SOH CAH TOA to help. And finally, substitute in the
values and solve the equation.
