In the given figure, the measure of
angle 𝐵𝐴𝐶 is equal to 90 degrees and the line 𝐴𝐷 is perpendicular to 𝐵𝐶. What is 𝐴𝐶 tan 𝜃?
So in this question, we’ve got a
right-angle triangle cause we’re told that the measure of angle 𝐵𝐴𝐶 is equal to
90 degrees. So that’s a right angle. So therefore, because we’ve got a
right-angle triangle, we know that we can have a look at the trigonometric
ratios. And when we’re dealing with the
trigonometric ratios, we have a memory aid, which is SOHCAHTOA. And what this does is it tells us
how to work out sin 𝜃, cos 𝜃, and tan of 𝜃.
In this question, we’re looking for
𝐴𝐶 tan 𝜃. Well therefore, we know that we’re
gonna be interested in TOA because that’s the one that deals with the tangent
ratio. And what TOA tells us that we said
is that tan 𝜃 is equal to the opposite divided by the adjacent. So what do we do now? Well, the next step is to label our
Well, the first side that we’re
gonna label is our hypotenuse. And that’s because this is the side
that’s opposite the right angle. And it’s also the longest side of
our triangle. And the triangle we’re interested
in is the triangle 𝐴𝐵𝐶. Now, the next side we’re gonna
label is the opposite because this is the side opposite the angle, which is 𝜃. And the final side is the
adjacent. And this is the side that’s next to
the angle 𝜃 and also between the angle 𝜃 and the right angle.
So in this question, we’re
interested in the opposite and the adjacent because we’re dealing with tan 𝜃. So therefore, if we substitute
these into our formula for tan 𝜃, we can see that tan 𝜃 is equal to 𝐴𝐵, the
opposite, divided by 𝐴𝐶, which is the adjacent. And then, what we can do is
multiply each side of the equation by 𝐴𝐶 to remove it from the denominator so to
get rid of the fraction.
And when we do that, we get 𝐴𝐶
tan 𝜃 is equal to 𝐴𝐵. So that solved the problem because
what we were looking for is what 𝐴𝐶 tan 𝜃 was. And 𝐴𝐶 tan 𝜃, as we’ve already
said, is 𝐴𝐵.