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 it’s definition then is that, for a particular angle 𝜃, the sine 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 a 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 the sine
button, there is, in fact, this sine inverse as well. That would 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
So here is our first question. We have a right-angled triangle in which we’re
given the length 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 sine 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 sine of 𝜃, which is unknown, is equal to nine over sixteen. 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 the inverse sine
function that we talked about. And this tells me that this angle 𝜃 is equal to the inverse sine of nine
over sixteen. 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 thirty-four point two two eight
eight six 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 was equal to thirty-four 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 sine 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 write down the Pythagorean theorem for this triangle. So it will tell
me that seven squared plus 𝑦 squared is equal to twenty-five 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 twenty-five squared with their
actual values. So I have that forty-nine plus 𝑦 squared is equal to six hundred and
twenty-five. Next, I need to subtract forty-nine from both sides of this equation. And doing so tells me that 𝑦 squared is equal to five hundred and seventy-six.
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 five hundred and seventy-six, and
that is equal to twenty-four. 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 twenty-four divided
by twenty-five. 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 nought point nine six. 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
ask us for the value of sine 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.
Out next problem is a worded problem. It tells us that a ramp is four meters
long and thirty centimeters 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
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 meters long. So this measurement here, for the
length of the ramp, is four meters. And we’re told it’s thirty centimeters high. Now we need
to be careful because those units are different. So I’m gonna convert that into meters. And
therefore, this length is zero point three meters.
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 could 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 sine 𝜃 is equal to zero point three 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 sine
of this ratio, naught point three 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 four point three zero one
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, four
point three, 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 then in 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 sine of the angle 𝜃 is
equal to five over eighteen. Now I want to work out this angle 𝜃, so I need to use that inverse sine
function. And therefore, I have that 𝜃 is equal to sine inverse of this ratio, five
Now I can evaluate that using my calculator. And this gives me that 𝜃 is equal to sixteen point one two seven six 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 sixteen full degrees, first of all. And then I have this decimal, zero
point one two seven six two and so on left over, which needs to be converted into minutes and
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 sixty. If I do this then, it gives me seven point six five seven two and so on. So that
tells me that there are seven full minutes. I’ve also then got this decimal of zero point six five seven two one left over.
And finally, this part I need to convert into seconds. So I have naught point six five seven minutes, and a second is one sixtieth of
a minute. So again, I then need to multiply this by sixty in order to work out how many
seconds this represents. So doing so gives me thirty-nine point four three two. So I have thirty-nine
seconds to the nearest second there. So pulling all of this together then, I have sixteen degrees, seven minutes,
thirty-nine seconds. And therefore, that’s my answer for the measure of angle 𝐴𝐵𝐶 to the
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.