Aya uses trigonometry to work out the size of angle 𝑥 in the following triangle. She assumes that angle 𝐴𝐵𝐶 is a right angle in her calculation. Angle 𝐴𝐵𝐶 is actually 88 degrees. How does this change her calculation? You must show your working.
So to solve this problem, what we need to do is work out the value of 𝑥 using the same assumption as Aya and then work out the value of 𝑥 using the actual value of angle 𝐴𝐵𝐶 which is 88 degrees. And then, we will compare them to see what changes there are in the calculation. So we’re gonna start by calculating the value of 𝑥 if angle 𝐴𝐵𝐶 was equal to 90 degrees. Well, if it was equal to 90 degrees, then we’d had a right-angled triangle, then we’d know that we’d be using either Pythagoras or trigonometry. But we’re told that Aya used trigonometry. So she’s gonna use the trig ratios.
And in order to calculate using the trig ratios, we have set steps that we’re gonna go through. The first step is to label the sides of our triangle. The first side we label is the hypotenuse because this is the longest side opposite the right angle and then the opposite is the side opposite the angle that we’ve got or the angle we’re looking for, which is 𝑥. And then, finally, the adjacent. It’s the side next to the angle or the side that’s between the angle and the right angle. That’s step one complete.
Now, we move on to step two. Well, step two is choose the ratio. So we need to choose which trigonometric ratio to use. And to do that, we have a memory aid to help us. And that is SOH CAH TOA. And this tells us that the sine of an angle is equal to the opposite divided by the hypotenuse. The cosine of an angle is equal to the adjacent divided by the hypotenuse. And the tangent of an angle is equal to the opposite divided by the adjacent.
So to enable us to choose which ratio to use, we’ve circled the sides that we’ve been given. So we’ve got the hypotenuse because that’s 6.5. And we got the adjacent because that’s five. So therefore, we’re gonna use the cosine ratio. And that’s because CAH has the adjacent and the hypotenuse or A and H. So therefore, we’ve completed step two and we’re gonna move on to step three.
Now, step three is to substitute. So what we need to do when we substitute is substitute the values we’ve got into the cosine ratio. So we’ve got cos 𝜃 is equal to the adjacent over the hypotenuse. So therefore, if we use the values we’ve got, we’ve got cos 𝑥 is equal to five over 6.5. So that step three complete. Let’s now move on to the final step, which is step four: solve.
So we now need to rearrange and solve to find 𝑥. Well, what we need to do to solve to find 𝑥 is take the inverse cosine of both sides of the equation. So when we do that, we get 𝑥 is equal to the inverse cosine of five over 6.5. A quick tip here I recommend that whenever you’re doing this kind of question, you actually put cosine to the minus one of five over 6.5 into the calculator. So don’t work it out first; put it all in together and that way you keep accuracy. And you find the inverse of cosine when you press shift and then the cosine button. And when we calculate this, we get 𝑥 is equal to 39.7. So now we’ve calculated 𝑥 if angle 𝐴𝐵𝐶 was equal to 90 degrees.
So now, what we gonna do is calculate 𝑥 if the angle 𝐴𝐵𝐶 was equal to 88 degrees. Now, because the angle 𝐴𝐵𝐶 is equal to 88 degrees, it means that we’ve no longer got a right angle in our triangle. So therefore, we can’t use the trig ratios. So now, we have to decide whether to use the sine rule or the cosine rule. Well, to decide what we need to do is see have we got a matching pair and the answer is yes. We’ve got 6.5 and 88 degrees. They’re called a matching pair because they’re an angle and the side opposite that angle.
So therefore, we’re gonna use the sine rule. And the sine rule tells us that sin 𝐴 over 𝑎 is equal to sin 𝐵 over 𝑏 is equal to sin 𝐶 over 𝑐. And we got it this way round because we’re looking to find an angle. So we’ve got the sin 𝐴, sin 𝐵, and sin 𝐶 as the numerators. It can be flipped the other way if you’re trying to find a side.
However, there seems to be a bit of a problem because to enable us to use the sine rule, we’d want a matching pair that’s gonna involve the angle we want, which is 𝑥. But we don’t know the side that’s opposite 𝑥. Is there anything we can do? Well, the answer is yes, there is because what we can do is we can find out the other angle, so the angle at 𝐴. And we can do that because then we’ll have three angles in the triangle and we’ll know that they add up to 180 degrees. So we can use that to find 𝑥. So we’ve now got our other matching pair.
So we’ve got our angle which I’m gonna call 𝑘 and then five because that’s the side opposite it. So therefore, using the sine rule, we can say that sin 88 over 6.5 is gonna be equal to sin 𝑘 over five. And then, we can multiply each side of the equation by five. And then, we get five sin 88 over 6.5 is equal to sin 𝑘. So now, we want to find what 𝑘 is. So again, we’re gonna use an inverse trig function. So this time, it’s gonna be the inverse of sin. So we’re gonna do sin to minus one of five sin 88 over 6.5. And this will give us 𝑘. So this gives us that 𝑘 is equal to 55.24 degrees.
So now, we have two angles in our triangle. So we can use the fact that we know that they sum to 180 to find 𝑥. So we can say that 𝑥 is gonna be equal to 180 minus then 88 plus 55.24 which gives us a value of 𝑥 of 41.76 degrees.
So therefore, we can say that if Aya uses an assumption that the angle 𝐴𝐵𝐶 is a right angle in her calculation, but the angle 𝐴𝐵𝐶 is actually 88, then Aya will calculate a value of 𝑥 which is less than the actual value and I’ve shown that with the workings out.