Video Transcript
In this video, we are going to see
how to use the sine ratio to calculate angles in right-angled triangles.
So, first of all, a reminder of
what the sine ratio is. I have here a diagram of a
right-angled triangle in which I’ve labelled one of the other angles as 𝜃. And then, I’ve labelled the three
sides in relation to this angle 𝜃. So, we have the opposite, the
adjacent, and the hypotenuse. Now, the sine ratio, remember, is
the ratio of the opposite and the hypotenuse in this triangle. So, its definition then is that,
for a particular angle 𝜃, the sin of 𝜃 is equal to the opposite divided by the
hypotenuse.
Now, that form of the ratio is very
useful if we’re looking to calculate the lengths of either the opposite or the
hypotenuse. But this video is about calculating
angles. And in order to do that, we need
another specification of this relationship. And this is involving what we refer
to as the inverse sine function. Now, the inverse sine function
basically works like this. It says, if I know the value of
that ratio, then I want to work backwards to tell me what is the angle 𝜃, to which
that ratio belongs.
So, it’s represented using this
notation here, sine and then a superscript negative one, which means the inverse
sine, or sine inverse. And so, we say that 𝜃 is equal to
sine inverse of the value of this ratio of the opposite divided by the
hypotenuse. If you look on your calculator,
you’ll often see that just above this sine button, there is, in fact, this sine
inverse as well. That will depend on your
calculator. But it’s usually just above the
button, and you have to press shift in order to get to it. So, now, we’ll see how we can use
this inverse sine ratio in order to calculate the size of an angle.
So, here is our first question.
We have a right-angled triangle
in which we’re given the lengths of two sides. And we’re asked to calculate
angle 𝜃 to the nearest degree.
So, as with any problem for
trigonometry, my first step is to label all three sides of the triangle in
relation to this angle 𝜃. So, we have the opposite, the
adjacent, and the hypotenuse. Now, looking at this triangle,
I can see that it’s the sine ratio that I need because I’ve been given the
lengths of the opposite and the hypotenuse. So, that is O and H, which is
the SOH part of SOHCAHTOA. Now, in this video, they’re all
going to be involving sine because that’s specifically what the video is
about. But in general, if you didn’t
know which of the ratios to use, that’s how you would work it out, by
determining which pair of sides is involved in the ratio.
So, I recall the definition of
sine, which is that sin of the angle 𝜃 is equal to the opposite divided by the
hypotenuse. And now, I’m gonna write down
this ratio for this particular triangle. So, I have that sin of 𝜃,
which is unknown, is equal to nine over 16. So, I know the value of the
ratio. And I want to work backwards in
order to find the angle that this ratio belongs to. So, this is why I’m going to
use that inverse sine function that we talked about.
And this tells me that this
angle 𝜃 is equal to the inverse sin of nine over 16. So, I need to use my calculator
to evaluate this. And remember, you may need to
use shift in order to get to that sine inverse button. But that will depend on your
calculator. And when I type this in, I will
get a value of 34.22886 for 𝜃. Now, I’m asked for 𝜃 to the
nearest degree. So, I need to round my
answer. And doing so then, tells me
that 𝜃 is equal to 34 degrees.
So, for this question then, we
identified that it was the sine ratio we needed because we had the opposite and
the hypotenuse. We wrote down the ratio for
this question. And then, we used the inverse
sine function in order to calculate this missing angle 𝜃.
The second question then, we’re
given a diagram of a right-angled triangle. And we’re asked this time, what
is the value of sin 𝜃. So, we’re just being asked to
write down the ratio, not actually work out the value of the angle.
So, first step, I’m gonna label
the three sides in relation to this angle 𝜃. And because I’m asked about the
sine ratio, I’m also gonna recall its definition. So, to answer this question,
what I need to do is just write down the value of this sine ratio, the opposite
divided by the hypotenuse. But looking at the diagram, I
can see that I haven’t actually been given the length of the opposite. I’ve been given the length of
the other two sides, the adjacent and hypotenuse.
But we can work out the length
of the opposite. It’s a right-angled
triangle. And I’ve been given the lengths
of two of the sides. So, you need to recall some
work from another area of mathematics. You need to recall the
Pythagorean theorem. Now, the Pythagorean theorem,
you’ll often see it written as 𝑎 squared plus 𝑏 squared is equal to 𝑐
squared. But the maths behind that, what
it tells us, is that if you have a right-angled triangle, if you take the two
shorter sides and square them and add it together, then you get the same result
as if you square the hypotenuse. So, what this enables me to do,
is calculate the length of the third side, if I know both of the other two.
So, I’m gonna use this
Pythagorean theorem in order to work out the length of the opposite. Now, I’m gonna give it a
different letter in order to save confusion with O being mixed up with zero, so
I’m gonna refer to it as 𝑦. So, now, I’ll write down the
Pythagorean theorem for this triangle. So, it will tell me that seven
squared plus 𝑦 squared is equal to 25 squared. And now, what I have is an
equation that I’ll be able to solve, in order to work out the value of 𝑦.
The first step is to replace
seven squared and 25 squared with their actual values. So, I have that 49 plus 𝑦
squared is equal to 625. Next, we need to subtract 49
from both sides of this equation. And doing so tells me that 𝑦
squared is equal to 576. So, next, to work out the value
of 𝑦, I need to find the square root of both sides of the equation. So, we have 𝑦 is equal to the
square root of 576. And that is equal to 24.
So, using the Pythagorean
theorem has enabled me to work out the length of this third side of the
triangle, the opposite. Therefore, I’ve got all the
information that I need in order to finish off the question. The question asked me to write
down the sine ratio, so it’s opposite divided by hypotenuse. And looking at the triangle, I
can see that that’s going to be 24 divided by 25. So, I could leave my answer
like that in a fractional form. Or in this case, I could
evaluate it as a decimal, which would be 0.96. So, either of those two formats
would be perfectly acceptable here.
Now, just a reminder, this
question doesn’t actually ask us to calculate the value of the angle. It just asks us for the value
of sin 𝜃. So, we’re okay to stop
here. If we were asked for the value
of 𝜃, we’d have to use that inverse sine function at this point here.
Our next problem is a worded
problem.
It tells us that a ramp is four
metres long and 30 centimetres high. In order for the ramp to be
safe for wheelchair users, the angle of inclination must be less than five
degrees. And we’re asked to determine,
is the ramp safe?
So, we haven’t been given a
diagram. And if you’re not given one, I
would always suggest you draw a diagram to start off with. So, we’re gonna have a diagram
of the floor, which is horizontal, the point where this ramp reaches, which is
vertical, and then the ramp itself. So, here is our diagram of that
ramp.
Now, we need to put the
information in the question on to it. We’re told that it’s four
metres long. So, this measurement here for
the length of the ramp is four metres. And we’re told it’s 30
centimetres high. Now, we need to be careful
because those units are different. So, I’m gonna convert that into
metres. And therefore, this length is
0.3 metres.
Now, we’re asked about the
angle of inclination. So, that’s the angle between
the ramp and the floor. It is this angle here. So, we can see that we have a
right-angled triangle. And therefore, this is a
problem that can be solved using trigonometry. As with all the previous
questions, I’m gonna start off by labelling the three sides of the triangle in
relation to that angle 𝜃. And we can see then that we’re
given the lengths of the opposite and the hypotenuse. So, we know that it’s the sine
ratio that we’re going to be using.
So, I’m gonna write out the
sine ratio using the information in the question. And that will tell me that sin
𝜃 is equal to 0.3 divided by four. Now, I want to work out the
value of 𝜃, this angle. So, I’m going to use the
inverse sine function. And what this would tell me is
that 𝜃 is equal to the inverse sin of this ratio, 0.3 over four, so as it’s
written on the screen here. Now, I can use my calculator to
evaluate this angle 𝜃 using that sine inverse button. And this tells me that this
angle 𝜃 is equal to 4.301 and so on.
Now, the question asked me, is
the ramp safe? And it tells me that it will be
safe if the angle of inclination is less than five degrees. So, as our value of 𝜃, 4.3, is
less than five, our answer to the question then is yes, this ramp is safe. So, for any worded problem, I
would always suggest drawing a diagram first. And then, from that point, it
just becomes very similar to other questions we’ve looked at. You label the sides, you write
down the sine ratio, and then use the inverse sine function in order to
calculate the angle you’re looking for.
Okay, our final question asks
us to calculate the measure of angle 𝐴𝐵𝐶, giving our answer to the nearest
second.
So, first of all, angle 𝐴𝐵𝐶,
that is the angle formed when we move from 𝐴 to 𝐵 to 𝐶, so it is this angle
here. Now, the question asks us for
our answer to the nearest second. So, we’ll have to recall how to
convert answers from degrees into degrees, minutes, and seconds later on.
Now, as this is a problem
involving trigonometry, I’m gonna start in the usual way. I’m going to label the three
sides of this triangle in relation to this angle 𝜃. So, I have their labels
here. And this confirms that it is
the sine ratio I’m going to need within this question because you’ll see I’ve
been given the lengths of the opposite and the hypotenuse. So, as in all the previous
questions, I’m now going to write this sine ratio out, substituting in the
information I know. So, I’m gonna have that sin of
the angle 𝜃 is equal to five over 18.
Now, I want to work out this
angle 𝜃, so I need to use the inverse sine function. And therefore, I have that 𝜃
is equal to sine inverse of this ratio, five over 18. Now, I can evaluate that using
my calculator. And this gives me that 𝜃 is
equal to 16.1276 and so on. Now, this answer is in degrees,
but the question has asked me to give my answer to the nearest second. So, I need to recall how to
convert an answer from degrees into degrees, minutes, and seconds.
So, I have 16 full degrees,
first of all. And then, I have this decimal,
0.12762 and so on left over, which needs to be converted into minutes and
seconds. Now, remember, a minute is one
sixtieth of a degree. So, in order to convert this
into how many full minutes there are, first of all, I need to multiply by
60. If I do this then, it gives me
7.6572 and so on. So, that tells me that there
are seven full minutes.
I’ve also then got this decimal
of 0.65721 left over. And finally, this part I need
to convert into seconds. So, I have 0.657 minutes. And a second is one sixtieth of
a minute. So again, I then need to
multiply this by 60 in order to work out how many seconds this represents. So, doing so gives me
39.432. So, I have 39 seconds, to the
nearest second there. So, pulling all of this
together then, I have 16 degrees, seven minutes, 39 seconds. And therefore, that’s my answer
for the measure of angle 𝐴𝐵𝐶 to the nearest second.
In summary then, we’ve reminded
ourselves of the definition of the sine ratio as the opposite divided by the
hypotenuse. We’ve seen how the inverse sine
function can be used to calculate the value of an angle. And then, we’ve applied it to
answer a couple of questions.
