### Video Transcript

In this video, we’re going to see
how to apply the inverse of the three trigonometric ratios, sine, cosine, and
tangent, in order to calculate angles in right-angled triangles.

First of all, let’s define these
inverse trigonometric ratios. I have here a diagram of a
right-angled triangle in which I’ve labeled one of the angles as 𝜃. I’ve then labeled the three sides
of the triangle in relation to the angle 𝜃, so we have the opposite, the adjacent,
and the hypotenuse. The three trigonometric ratios,
sine, cosine, and tangent, are the ratios that exist between different pairs of
sides in this right-angled triangle.

So, the sine ratio, sin of angle
𝜃, is the opposite divided by the hypotenuse. Cosine ratio is cos of 𝜃 is the
adjacent divided by the hypotenuse. And finally, tangent, the tan
ratio, is the opposite divided by the adjacent. A useful way to remember these is
to recall the word SOHCAHTOA, where each of those letters represents the first
letter in each of these words. So, CAH, for example, the C is for
cos, the A is for adjacent, and the H is for hypotenuse. So, CAH tells us that the cos ratio
is adjacent divided by hypotenuse.

Those ratios are the trigonometric
ratios, and they’re particularly useful if we know a side and an angle and are
looking to calculate another side. But in this video, we’re looking at
how to calculate angles. And therefore, we need what’s
referred to as the inverse trigonometric ratios. These are defined like this. The notation we use is sin, cos, or
tan and then a superscript negative one, which is said as sine inverse or inverse
sine. And what they mean are, if I know
the value of the ratio of opposite divided by hypotenuse, then I can work backwards
using this inverse sine function in order to calculate the angle that that ratio is
associated with.

So, when we know two sides of a
right-angled triangle, we can use the relevant inverse trigonometric ratio in order
to calculate an angle. On your calculator, you will see
that above the sin, cos, and tan button, there is usually sin inverse, cos inverse,
and tan inverse. You often have to press shift in
order to get to these functions. We’ll now look at a couple of
examples of how to apply these inverse trigonometric ratios.

For the given figure, find the
measure of angle 𝜃 in degrees to two decimal places.

So, we have a diagram of a
right-angled triangle, and we can see that we’re given the lengths of two of the
sides of this triangle. They’re three units and eight
units. And we’re looking to find the
size of this angle 𝜃.

As we’re using trigonometry for
this problem, the first step is going to be to label all three of the sides of
the triangle in relation to their angle 𝜃. So, we have the opposite, the
adjacent, and the hypotenuse. We can see then that the two
sides of the triangle we have are the adjacent and the hypotenuse. If I think back to the acronym
of SOHCAHTOA, then A and H appear together in the CAH part, which tells me that
it’s the cosine ratio I’m going to need to use in this question.

The definition of the cosine
ratio, remember, was that cos of the angle 𝜃 is equal to the adjacent divided
by the hypotenuse. So, I’m gonna write down this
ratio using the information in this particular question. And therefore, I have that cos
of 𝜃 is equal to three over eight. Now, this is where I need to
use the inverse trigonometric function. If cos of 𝜃 is equal to three
over eight, then 𝜃 is equal to cos inverse of three over eight.

At this stage, I use my
calculator to evaluate this, remembering that that cos inverse button is usually
directly above the cos button. This tells me that 𝜃 is equal
to 67.97568 degrees. The question asked me to round
my answer to two decimal places. Therefore, my final answer is
that 𝜃 is equal to 67.98 degrees. So, in this question, we
identified the need for the cosine ratio because the lengths we were given were
the adjacent and the hypotenuse. We wrote down that ratio using
these lengths. We then used the inverse cosine
ratio in order to calculate the value of this angle 𝜃.

Find the measure of angle ACB
giving the answer to the nearest second.

I’ve got a diagram of a
right-angled triangle, and I’m asked to find the measure of angle ACB. So, that means the angle formed
when I move from A to C to B. It’s this angle here that I’m
looking for, so I’ve given it the label 𝜃. As we’re going to use
trigonometry to solve this problem, I’m gonna label the three sides of the
triangle with their names in relation to that angle of 𝜃. So, we have the opposite, the
adjacent, and the hypotenuse.

Now, I can see that the two
sides I’ve been given are the opposite and the adjacent. If I think back to that acronym
SOHCAHTOA, then I see that it’s the tan ratio I’m going to need because the
opposite and the adjacent appear together in the TOA part. So, the definition of the tan
ratio is the opposite divided by the adjacent. For this triangle, then, that
is 43 divided by 26. So, we have tan of 𝜃 is equal
to 43 over 26. Now, I need to use the inverse
tangent ratio.

So, if tan 𝜃 is 43 over 26,
then 𝜃 is equal to tan inverse of 43 over 26. I use my calculator to evaluate
this. And it tells me that 𝜃 is
equal to 58.84069. Now, this is an answer in
degrees, and the question has asked me to give my answer to the nearest
second. So, I need to recall how to
convert a value from degrees into degrees, minutes, and seconds. So, I have 58 full degrees, and
then I have this decimal of 0.840695 and so on, which needs to be converted into
minutes and then seconds.

Remember that a minute is one
sixtieth of a degree. So, to work out what this
decimal represents in minutes, I need to multiply it by 60. When I do that, I get this
value of 50.4417295. This tells me that there are 50
full minutes and a decimal of 0.4417295, which needs to be converted into
seconds. A second is one sixtieth of a
minute. So, again, to convert this into
seconds, I need to multiply it by 60. When I do this, I get a value
of 26.5037. So, in order to round this to
the nearest second, I round it up to 27 seconds.

Finally, I need to pull the
three parts of this answer together. And in doing so, my final
answer then is that the measure of angle ACB is 58 degrees, 50 minutes, 27
seconds to the nearest second. So, within this question, we
identified the need for the tan ratio, because the lengths we were given were
the opposite and the adjacent. We wrote down the tan ratio for
this triangle. We then applied the inverse tan
ratio to work out the angle that this ratio was associated with and finally
converted this answer from degrees into degrees, minutes, and seconds.

A car is going down a ramp
which is 10 metres high and 71 metres long. Find the angle between the ramp
and the horizontal, giving the answer to the nearest second.

So, this question is a worded
problem, and we haven’t been given a diagram. I would always suggest that, in
this situation, you draw your own diagram to start off with. So, here we have an idea of
what this ramp might look like. It is 71 metres long and 10
metres high. We’re asked to find the angle
between the ramp and the horizontal, so we’re looking to calculate this angle
here, which I’ve labeled as 𝜃.

Now, we’re going to solve this
problem using trigonometry. So, I’m going to begin by
labeling the three sides of this triangle in relation to this angle 𝜃. Having done this, I can see
that the two lengths that I’ve been given represent the opposite and the
hypotenuse of this right-angled triangle. Thinking back to that acronym
of SOHCAHTOA, this tells me that it’s the sine ratio I’m going to need in this
problem, as O and H appear together in the SOH part of SOHCAHTOA.

So, the definition of the sine
ratio is that it’s the opposite divided by the hypotenuse. I then write this ratio down
for this particular triangle, and I have then that sin of 𝜃 is equal to 10 over
71. In order to work out the size
of this angle, I need to apply the inverse sine ratio. So, if sin of 𝜃 is 10 over 71,
then 𝜃 is sin inverse of 10 over 71. I evaluate this with my
calculator, and I see that 𝜃 is equal to 8.09674977.

Now, this is an answer in
degrees, and the question has asked me to give my answer to the nearest
second. So, let’s recall how to convert
an answer from degrees into degrees, minutes, and seconds. I have eight full degrees, and
then I have a decimal left over of 0.09674977, which needs to be converted first
into minutes and then into seconds. Now, recall that a minute is
one sixtieth of a degree. So, in order to work out what
this decimal represents in minutes, I need to multiply it by 60. This gives me a value of
5.804986188.

What this tells me then is that
I have five full minutes and then a decimal of 0.80 and so on left over. This decimal needs to be
converted into seconds. So, recall that a second is one
sixtieth of a minute. And therefore, in order to see
what this decimal represents in seconds, I need to multiply it by 60. When I do this, I get a value
of 48.29917126. And if I round that to the
nearest second, it’s 48 seconds. Finally, then, I need to
combine the three parts of my answer. And this tells me that the
angle between the ramp and the horizontal, to the nearest second, is eight
degrees, five minutes, and 48 seconds.

So, within this question, we
drew our own diagram as we weren’t given one within the question itself. We identified the need for the
sine ratio because the two lengths we have been given were the lengths of the
opposite and the hypotenuse. We used the inverse sine ratio
to calculate the angle that was associated with that value. And then, we converted our
answer from degrees into degrees, minutes, and seconds.

In summary, then, the three inverse
trigonometric ratios sine inverse, cosine inverse, and tan inverse can be used to
calculate an angle in a right-angled triangle when we know at least two of the
sides. The ratio that we choose will
depend on which two sides we’re given in exactly the same way as it does when we’re
calculating the length.