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Video: Finding Angles Using the Sine Ratio

Lauren McNaughten

Learn how to use the inverse sine ratio to calculate an angle in a right triangle. This includes the use of the Pythagorean theorem to first calculate a side length when the sides given are not the required lengths for use in the sine ratio.

12:52

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 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 angle.

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 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 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 over eighteen.

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 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 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 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.