𝐴𝐵𝐶 is a right-angled
triangle. Work out the size of angle 𝑥. Give your answer to one decimal
So the first thing to do when we’re
solving a problem like this is think, okay, what are we gonna use to help us do
this? So we can see that it’s a
right-angled triangle. So instantly, we go okay, we can
use Pythagoras or can use the trig ratios.
Next, we have a look at our
triangle. And we can see that, actually, what
we’ve got here is an angle. So we’re trying to find an
angle. So therefore, we know that we’re
actually gonna use our trig ratios, because although we’ve got both sides 𝐴𝐶 and
𝐶𝐵, so therefore we could find 𝐴𝐵 using Pythagoras. The question is actually asking us
to find an angle. So we know that it’s gonna be using
the trig ratios.
Now whenever solving a problem like
this using our trig ratios, I like to follow some basic steps to actually help us
make sure that we actually don’t miss out and make any mistakes.
So step one is actually labelling
the sides. So the first side that I’ve
actually labelled is 𝐻 or the hypotenuse. And this is because it’s the
longest side and opposite the right angle. The next side that I’ve actually
labelled is the opposite. And that’s 𝐴𝐶, because it’s
actually opposite the angle that we’re looking for. And then the final side that I’ve
actually labelled is 𝐴, which is 𝐶𝐵. And that’s because that’s the
adjacent. And that’s actually adjacent to the
angle we’re looking for. Okay, great! So that’s step one complete.
So now step two is to actually
choose the ratios. We need to decide which one of our
trig ratios we’re actually gonna use to solve the problem. So to actually help us to remember
what to do with this, we actually use SOHCAHTOA, because this can actually help us
remember our trig ratios.
So we’ve got SOH, which means that
the sine of an angle is equal to the opposite divided by the hypotenuse; CAH, which
means that the cosine of an angle is equal to the adjacent over the hypotenuse; and
TOA, which means the tan of an angle is equal to the opposite over the adjacent.
Okay, so what we need to do is
actually choose which one of these ratios we’re going to use. Well, what I’ve actually done is
circle the sides that we have. So we have the opposite and the
adjacent. So then if we go back to our
SOHCAHTOA, we can see, well, it’s TOA that actually has the opposite and the
adjacent. So we know that we’re gonna use the
So just to remind us, we’ve got tan
𝜃 is equal to the opposite divided by the adjacent. So then what we’re gonna do is move
on to step three. And step three is to actually
substitute in our values. So we’ve chosen our ratio. So now we’re gonna substitute in
our values. So we get tan 𝑥 is equal to 21,
our opposite, divided by 26, which is our adjacent.
So now what we do is we actually
move on to our final step, which is step four, which is solve. And the first thing we’re gonna do
to actually solve this equation is actually take the inverse tangent or inverse tan
of each side of our equation. And when we do that, we get 𝑥 is
equal to tan to minus one or inverse tan of 21 over 26.
Well, at this point is to make sure
that when you’re actually using a calculator, you press the shift tan button. And that’s how you’d find the
inverse of tan. So you press shift and then press
tan. Also, another quick thing to
actually note here is actually find it easier to put tan to minus one or shift tan
of 21 over 26. Don’t try and work out separately
because then you might have some errors that are caused by inaccuracy when
Okay, when we actually calculate
this, we get 𝑥 is equal to 38.92754 et cetera degrees. Okay, so if you haven’t got this,
you’ll ask yourself why haven’t I got this? I’ve done anything else right. Why haven’t I got this right
answer? And that’ll be a very common
mistake. And that common mistake is actually
that the calculator is in the wrong setting. So you need to make sure that, in
the settings, so in the little window we have you display there’s either a d or a
capital D or deg, something to noting that actually your calculator is in degrees
not radians or anything else.
Okay, so once you’ve done that then
you’ll know you’ll get this answer. And so that’s why I said it’s
really really important if you’re ever doing any questions or going into an exam,
make sure you know your calculator and how it works.
Okay, so now we look back at the
question. How do we want our answer left? Well, we can see from the question
that we want our answer left to one decimal place. So therefore, we can say that the
size of angle 𝑥 is equal to 38.9 degrees to one decimal place.