Well yes it is true the sine rule is not just about finding length and distances it is also
about uncovering angles.
In this example we are looking for an angle and the angle that we are looking for is down
here pronumeral capital Z, remember the capitals refer to the angles and little letters refer
to the length.
So what do we have here?
We have this triangle here and we are given 3 pieces of information we are looking for
1, missing 1 as I have said the missing angle right down here.
Now what do we have?
We do have another angle here it is a 122 degrees it is angle Y an opposite directly
opposite that is-- the length 39.7cm, now the angle Z that we are looking for directly
opposite-- the length 27.5cm we are going to take this pieces of information and install
them into this formula up here this sine rule which you have done before so let us put them
in there.
So taking a look of what we have, we have a straight away, we have 39.7cm opposite to
Y, so let us start up for the Y. little y sine/ sinY, capital of Y is the angle, so
let us put them in here so, little y 39.7cm, let us put that in there 39.7/ sinY, which
is the angle down here directly opposite can you see that yeah, 122degrees now that is
going to = zed right?
That is the distance zed which is right here 27.5 let us put that in thereóall over the
angle directly in front the missing angle sin zed-- Ok this our pieces of information
3 unknown, 1 unknown the angle up there is what we are looking for so what do we do?
As always, we cross multiply so let us do that, let us cross them, multiply all the
way across.
So we will haveósin Z x 39.7 = that is going to be 27.5 x sin 122.
Ok what do we want to get by itself it is right here the ZED the little zed here that
is the pronumeral particularly right there, that is what we are looking for, so let us
divide both sides by 39.7, so let us do that /39.7 and divide by 39.7.
as you can see here we can cancel the 39.7 so let us do that I am going to cancel it
now, cancel, cancel, we are left with sinZ = 27.5 x sin122 / 39.7, let us put this stuff
in the calculator right now.
so, we are going to get ñ now on the calculator 27.5 x sin122 that is going to yield 23 that
is going to yield 23.32/ what we have there which is the 39.7 , ok let us put that in
the calculator again.
Ah. 23.32 /39.7 is going to give us an = 0.5874 we will just going to take it 4 decimals places
and then we are going to find Z now.
To find Z right here sin Z we are going to shift Z on that so if you calculate defending
what you have shift Z, sin Z or 2nd function or inverse whatever you have on your calculator
so we are looking for Z, what do we got the angle right here Z is what we are looking
for so capital Z which is the angle = you got shift sin0.5874 that is going to give
us to the nearest degree I am going to round that up is ñ 36degrees.
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