Video: Finding the Unknown Angles in a Right Triangle Using Trigonometry

For the given figure, find the measures of ∠𝐴𝐵𝐶 and ∠𝐴𝐶𝐵, in degrees, to two decimal places.

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Video Transcript

For the given figure, find the measures of angle 𝐴𝐵𝐶 and angle 𝐴𝐶𝐵, in degrees, to two decimal places.

There are two parts in this question, finding the measure of angle 𝐴𝐵𝐶 and finding the measure of angle 𝐴𝐶𝐵. And they’re both gonna involve four steps. And we’ll start with finding the angle of 𝐴𝐵𝐶. Well, first of all, because we’re using a right-angled triangle and we’re given two sides and we want to find a different angle, then therefore we know that we can use the trigonometric ratios. And that’s what we’re gonna use to solve this.

To help us understand step one, what I’ve also done is I’ve actually marked the angle we’re looking for. So that’s angle 𝐴𝐵𝐶 on our diagram. So for step one, we’re just gonna label our sides. So we start with our hypotenuse, which is opposite the right angle and also the longest side. Next, we have the opposite, which is opposite the angle we’re looking for. So that’s opposite 𝐴𝐵𝐶. And then finally, we have the adjacent. And the adjacent is the angle that’s next to the angle that we’re looking for. Okay, great. So we’ve now completed step one and we’ve labeled the sides.

Now we’re gonna go on to step two. And step two is we actually have to choose the ratios. We have to choose which of our trigonometric ratios we’re gonna use to solve the problem. To help us to do that, what I have is this mnemonic SOH CAH TOA cause it helps me remember which one of the trigonometric ratios I’m going to use. Okay, so now let’s have a look at which sides we have. So in this problem, we have the adjacent. And we have the hypotenuse. So therefore, I can go back to my SOH CAH TOA. And I have a look. And I see it, right. Which part has the adjacent and the hypotenuse? And we can see it’s the middle part. So therefore, we know that we’re gonna use the cosine ratio. And we know from SOH CAH TOA that the cosine of an angle is gonna be equal to the adjacent divided by the hypotenuse. Okay, great. Let’s move on to step three.

So step three is actually to substitute the values we have into our equation. So therefore, we can say that the cosine of angle 𝐴𝐵𝐶 is equal to four over nine. So great. We’ve completed step three. Let’s move on to the final step.

And the final step is to rearrange and solve. And to enable us to solve this, what we’re actually gonna do is we need to take the inverse cosine of both sides. So therefore, we can say that the angle 𝐴𝐵𝐶 is equal to the inverse cosine of four over nine. And this gives us that the angle 𝐴𝐵𝐶 is equal to 63.6122. And if we look back at the question, we can see that the question wants us to leave the answer to two decimal places. So therefore, angle 𝐴𝐵𝐶 is equal to 63.61 degrees, to two decimal places. Okay, great. So we’ve now found angle 𝐴𝐵𝐶. Let’s move on to angle 𝐴𝐶𝐵.

Right. Now to find angle 𝐴𝐶𝐵, we’re gonna complete all four steps again. So step one, let’s label our sides. Well, first of all, we have the hypotenuse. And this doesn’t change because this is still opposite the right angle, still the longest side. And then next, we have the opposite. And this time, the opposite is actually 𝐴𝐵 not 𝐴𝐶. And that’s because it’s opposite our angle which we’ve remarked here in pink which is angle 𝐴𝐶𝐵. And then finally, I’ve marked on the adjacent. Okay, great. Step one, sides labelled. Okay, we’re gonna move on to step two.

We now need to choose our ratio. This time, we actually have the hypotenuse and the opposite because as we said, the opposite and adjacent have changed because we’re looking at a different angle this time. So therefore, I can look at my mnemonic SOH CAH TOA. And I can see that the opposite and the hypotenuse are both involved in the first part which is SOH. So therefore, I can see that we’re gonna use the sine ratio. And from our mnemonic, we can see that the sine ratio is going to be equal to the opposite divided by hypotenuse. Okay, great. Step two, done. We’ve chosen our ratio. So on to step three.

Let’s substitute in our values. So therefore, we can say that the sine of angle 𝐴𝐶𝐵 is gonna be equal to four over nine. Okay, that’s because four is our opposite and nine is our hypotenuse. Okay, great. So that’s step three completed. Let’s move on to step four which is rearrange and solve.

So now we’re gonna rearrange and solve to find out what angle 𝐴𝐶𝐵 is. This time, we’re gonna take the inverse sine of both sides of our equation. So therefore, we’re gonna get that the angle 𝐴𝐶𝐵 is equal to the inverse sine of four over nine. So a quick tip at this point, if you’re not sure where to find the inverse sine on your calculator, some people call it shift sine. And the reason they say that is because if on your calculator you find the shift button, you press the shift button, and then press sine, usually that will help you find your inverse sine.

Okay, great. So now let’s solve this and find out what angle 𝐴𝐶𝐵 is. Okay, great. So when we solve this, we get 26.3877, et cetera. So again, we want to leave our answer to two decimal places. So let’s round this which gives us 26.39 degrees, to two decimal places. So fantastic! We can say that for the given figure, the measures of angle 𝐴𝐵𝐶 and 𝐴𝐶𝐵, in degrees, to two decimal places, are 63.61 and 26.39, respectively.

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