Video: Using Properties of the 45-45-90 Degree Triangle

Learn the exact sine, cosine and tangent ratios for a 45° angle. Apply these exact ratio values to calculate lengths of sides in right-angled triangles with angles of 45° given.

11:16

Video Transcript

In this video, we’re going to see how the three trigonometric ratios, sine, cosine, and tangent, can be expressed exactly in the form of surds for an angle of 45 degrees. We’ll also see how to apply these ratios to some problems.

We’re going to begin then with a right-angled triangle. And it’s a special type of right-angled triangle because it’s isosceles. So, the two shorter sides are both the same length. And for ease, we’ll say that they’re both one unit. Now, because it’s isosceles, that also means that the other two angles in this right-angled triangle must be the same as each other. We’ve already used 90 degrees for the right angle, so the remaining 90 degrees needs to be split in half. And therefore, the other two angles are each 45 degrees.

So, we have what’s referred to as the 45-45-90-degree triangle. Now, we can also work out the third length of this triangle using the Pythagorean theorem. Remember, the Pythagorean theorem tells us that if you take the two shorter sides of a right-angled triangle and square them and add them together, we get the same result as if we square the longest side. So, if I give that longest side, the hypotenuse, a letter 𝑥, then the Pythagorean theorem tells me that 𝑥 squared is equal to one squared plus one squared.

I have then that 𝑥 squared is equal to two. And then, by square rooting, I see that the exact value of this side 𝑥 is root two units. So, I have the length of all three sides of this triangle. Now, what I want to do is work out the sine, cosine, and tangent ratios for an angle of 45 degrees. Now, for simplicity, I’m just going to use this angle here. And I want to label the three sides of this triangle with their names in relation to this angle of 45 degrees. So, I have the opposite, the adjacent, and the hypotenuse.

Now, I’m going to write down the three trigonometric ratios. So, sine, first of all, sine, remember, is equal to the opposite divided by the hypotenuse. So, using the values for this triangle, I have then sin of 45 is equal to one over root two. Now, I don’t want to leave it like that because that currently has a surd in the denominator. So, I’d like to rationalize this by multiplying it by root two over root two. That doesn’t change the value, of course, because root two over root two is just equivalent to one, but it gives me a value where the denominator has been rationalized.

And the value that it gives me is just root two over two. So, this is the exact value of sin of 45 degrees. I want to express it using a surd rather than a decimal because if I converted it to a decimal, it would need rounding and, therefore, would lose some of its accuracy.

Right, now, let’s do the same thing for cos of 45. So, cos, remember, is the adjacent divided by the hypotenuse. And looking at the triangle, I can see that that’s going to be one over root two again. As before, then, I would then rationalize this value. So, cos of 45 is also equal to root two over two. Finally, I want to look at the tangent ratio. Tan, remember, is the opposite divided by the adjacent. So, looking at the triangle, that’s one divided by one cause, remember, this was an isosceles triangle. And, therefore, that just simplifies to one. So, the exact value of tan 45 is one.

So, why are these important? Well, as I’ve already said, these are exact values in terms of surds, whereas if I were to convert them to decimals, they would need to be rounded in some way. So, working with an angle of 45 degrees enables me to give exact values as my answers. Secondly, if I can remember these values and learn them off by heart, then I can do trigonometry when I haven’t got a calculator if it’s an angle of 45 degrees within the question. So, you do need to learn these values and you need to be able to recall them. We’ll now look at how to apply this then to a couple of problems.

This problem is a worded problem. It tells us, a ladder eight metres long leans against a wall. We’re asked to find the horizontal length between the base of the ladder and the wall, given that the angle between the ladder and the ground is 45 degrees.

So, we haven’t been given a diagram for this problem. And I would always suggest that your first step is to draw your own. So, we’ll start off with a diagram of a ladder, a wall, and the ground. So, the ladder, the wall, and the ground form a right-angled triangle. And we’re told the ladder is eight metres long, so I’ve put that on. And we’re told the angle between the ladder and the ground is 45 degrees. So, my diagram looks like this.

The length I’m looking to find is that horizontal distance between the base of the ladder and the wall, so I’ve given it the letter 𝑥. Right, now, I’ve got my diagram. The next step is going to be to label the three sides of this triangle in relation to that angle of 45 degrees. And doing that, I have the opposite, the adjacent, and the hypotenuse. Now, this shows me that it’s the cosine ratio that I’m going to need because I want to work out the adjacent, and I’m given the hypotenuse. So, A and H appear together in the cosine ratio, the CAH part of SOHCAHTOA.

The definition of the cosine ratio, remember, is that cos of the angle 𝜃 is equal to the adjacent divided by the hypotenuse. So, looking at this triangle, I’m now going to write out the cosine ratio using the information in this question. I have then that cos of 45 is equal to 𝑥 over eight. This is an equation that I want to solve in order to work out the value of 𝑥. So, 𝑥 is currently divided by eight, which means I need to multiply both sides of this equation by eight. And this gives me that 𝑥 is equal to eight cos 45.

Now, this is where the size of that angle is significant. 45 degrees is one of those special angles for which we need to know the exact values of the sine, cosine, and tangent ratios. Cos of 45 can be written down exactly in terms of surds. And if you recall, cos of 45 is equal to root two over two. It’s perfectly possible that you could be asked this question or one like it in a setting where you don’t have access to a calculator because you need to remember the value of cos of 45.

So, we’ve recalled it here. And now we can just substitute this value directly into our calculation. So, we have then that 𝑥 is equal to eight multiplied by root two over two. And then, that just simplifies to four root two. So, our answer to the question then, with the units included, is that the distance between the base of the ladder and the wall is four root two metres. And that’s an exact value. We haven’t used a calculator at any point in order to approximate it.

So, within this question, we identified the need for the cosine ratio because it was the adjacent and the hypotenuse sides that were involved. And then, because it was an angle of 45 degrees, we recalled the exact value of the cosine ratio for 45 degrees in terms of a surd and used that exact value in our working out.

In this question, we’re given a diagram of a right-angled triangle and we’re asked to calculate the length of 𝐴𝐵.

Now, looking at the diagram, we can see we’re given one length. We can see the hypotenuse is 10 centimetres. But we’re not given the length of either the other two sides or indeed any of the angles apart from the right angle. We can, however, work out some information about this triangle because those two lines on sides 𝐴𝐵 and 𝐵𝐶 indicate that those sides are the same length. And therefore, this is an isosceles triangle.

As it’s an isosceles triangle, we can also see that the sizes of the other two angles, they must both be 45 degrees because they’re half of the 90 degrees that is left over. What this means then is that we have a 45-45-90-degree triangle. And therefore, we can answer this problem using the exact trigonometric values for an angle of 45 degrees. Incidentally, you could also answer this problem using the Pythagorean theorem, but that’s an alternative approach.

So, as we’re going to do it using trigonometry, I’m going to begin by labeling the three sides of this triangle in relation to one of those 45-degree angles. And I’m going to use this angle here. So, in relation to the 45-degree angle I’ve marked in green, I have the opposite, the adjacent, and the hypotenuse. Now, doing that enables me to see that it’s the sine ratio I’m going to use because I want to calculate 𝐴𝐵, which is the opposite, and I know the length of the hypotenuse. So, O and H appear in the sine ratio.

So, I recall the definition of the sine ratio. And now, I’d like to write it down for this triangle specifically. So, I’m gonna replace the angle with 45, the opposite is unknown, and the hypotenuse is 10. So, I have then that sin of 45 is equal to 𝐴𝐵 over 10. Now, I want to solve this equation in order to calculate 𝐴𝐵. So, I’m gonna multiply both sides of the equation by 10. In doing that then, I have that 𝐴𝐵 is equal to 10 sin 45.

Now, suppose you haven’t got a calculator for this question. 45 degrees, remember, is one of those special angles for which we need to be able to recall the exact values of the sine, cosine, and tangent ratios. And indeed, if you remember, sin of 45 is equal to root two over two. So, I can recall this value exactly, and then I can just substitute it directly into my calculation of 𝐴𝐵. So, I have then that 𝐴𝐵 is equal to 10 multiplied by root two over two. And that will simplify then to five root two. So, within this question, because I was able to express sin 45 exactly using a surd, I was also able to give my answer for 𝐴𝐵 in an exact format, five root two, as opposed to a rounded decimal.

In summary, then, we’ve learnt the exact values of the three trigonometric ratios, sine, cosine, and tangent, for an angle of 45 degrees. We’ve seen where they come from by starting with an isosceles right-angled triangle, where the two shorter sides are both of length one unit. We’ve then seen how to apply these ratios in questions where we can give an exact answer in terms of a surd.

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