0:00:01.330,0:00:05.609
There are 6 addition formula.[br]I'm not going to prove them.

0:00:06.410,0:00:09.578
But I am going to do is start[br]with one of them.

0:00:10.210,0:00:12.100
And derive a second one.

0:00:12.750,0:00:17.443
Then I'm going to take another[br]one is given and derive a second

0:00:17.443,0:00:22.136
one from that, and then we're[br]going to use those four to help

0:00:22.136,0:00:27.190
us derive the final two. So this[br]is the one we're going to start

0:00:27.190,0:00:31.112
with. The sign of a

0:00:31.112,0:00:36.360
plus B. This is where they get[br]their name from addition formula

0:00:36.360,0:00:40.637
because here we have a sub and[br]we're going to find it sign.

0:00:41.260,0:00:48.070
This breaks down.[br]Assign a Cosby

0:00:48.680,0:00:52.064
Plus calls a

0:00:52.064,0:00:58.387
sign be. Notice in terms[br]of remembering it, we keep the

0:00:58.387,0:01:03.103
A&B in the same order and the[br]signs and the cosines alternate.

0:01:04.100,0:01:10.644
So what about the[br]sign of a minus

0:01:10.644,0:01:17.980
B? Well, I just have[br]a think about this minus B.

0:01:18.520,0:01:25.820
It can be the sign[br]of a plus minus B.

0:01:27.260,0:01:30.860
And now. The formula that we

0:01:30.860,0:01:36.102
had here. Can be exactly the[br]same As for this one.

0:01:37.010,0:01:40.405
But in this one we can replace

0:01:40.405,0:01:47.518
B. By minus be[br]so let's do that sign.

0:01:48.140,0:01:54.500
A. Calls[br]of minus B.

0:01:55.130,0:01:58.580
Plus calls a.

0:01:59.270,0:02:02.450
Sign of minus

0:02:02.450,0:02:09.180
B. Well, the[br]sign is OK.

0:02:09.180,0:02:14.390
And the cause of minus B is[br]just cause big.

0:02:16.550,0:02:23.158
The cause a is OK, but the sign[br]of minus B is minus sign B, so

0:02:23.158,0:02:28.940
that's going to change that plus[br]sign into a minus sign and I can

0:02:28.940,0:02:31.831
just write sign be at the end.

0:02:32.940,0:02:35.988
So that's the second.

0:02:36.940,0:02:40.615
Of our addition

0:02:40.615,0:02:46.878
formula. If we've got these[br]for sign, it seemed reasonable

0:02:46.878,0:02:52.180
to expect that will have the[br]same things for cause for

0:02:52.180,0:02:56.518
cosine, so let's have a look[br]cause of A+B.

0:02:57.110,0:03:04.090
Well, this is cause a[br]caused B minus sign, a

0:03:04.090,0:03:11.070
sign be and that's our[br]starting one. So let's do

0:03:11.070,0:03:17.352
the same as we did[br]here cause of A-B.

0:03:18.430,0:03:25.550
Will rewrite the minus B[br]as a plus minus B.

0:03:26.370,0:03:32.519
And then we can replace the bees[br]in here in this first formula

0:03:32.519,0:03:37.249
with the minus be there so will[br]have cause a.

0:03:38.040,0:03:41.408
Cause of minus B.

0:03:42.280,0:03:49.175
Minus sign, a sign[br]of minus B.

0:03:50.820,0:03:58.254
Now the cause of minus B is[br]just cause be, so we have caused

0:03:58.254,0:03:59.847
a caused B.

0:04:01.300,0:04:07.176
And then we have minus sign a[br]Times by sign of minus B.

0:04:07.930,0:04:13.420
But the sign of minus B is a[br]minus sign be, so we have two

0:04:13.420,0:04:17.446
minus signs together, giving us[br]a plus sign plus sign a.

0:04:17.950,0:04:19.950
Sign be.

0:04:21.270,0:04:26.982
So we've now got 4 addition[br]formula. Let me turn over and

0:04:26.982,0:04:29.838
write those down as a group.

0:04:31.090,0:04:34.510
Sign of A+B?

0:04:35.520,0:04:37.839
Is sign a?

0:04:38.460,0:04:44.949
Call speedy close calls[br]a sign be.

0:04:45.700,0:04:49.940
Sign of a minus[br]B.

0:04:50.990,0:04:57.770
Is sign a[br]Cosby minus cause

0:04:57.770,0:05:01.160
a sign be?

0:05:01.160,0:05:04.800
Cause of

0:05:04.800,0:05:10.520
A+B? Is cause[br]a calls B?

0:05:11.050,0:05:14.950
Minus sign a

0:05:14.950,0:05:22.500
sign B. The[br]cause of A-B is

0:05:22.500,0:05:30.372
cause a caused B[br]plus sign a sign.

0:05:31.060,0:05:33.450
So there are our four.

0:05:34.230,0:05:38.622
Addition formula. Now we did[br]promise 6 but these are the fall

0:05:38.622,0:05:42.648
really basic ones. And really[br]these are the four that you've

0:05:42.648,0:05:43.746
got to learn.

0:05:44.450,0:05:47.590
The other two will presumably

0:05:47.590,0:05:51.081
be. Tangent tan of

0:05:51.081,0:05:58.159
A+B? Well, we[br]can derive these from the

0:05:58.159,0:06:05.829
others. Tan is sign over[br]cause so we have the

0:06:05.829,0:06:12.732
sign of A+B divided by[br]the cost of A+B.

0:06:13.770,0:06:18.858
So we can now make the[br]replacement for sign of A+B by

0:06:18.858,0:06:20.130
its expansion here.

0:06:20.670,0:06:26.038
And cause of A+B by[br]its expansion here?

0:06:26.040,0:06:29.220
Sign a caused

0:06:29.220,0:06:33.028
B. Close calls a

0:06:33.028,0:06:38.190
sign be. All[br]over.

0:06:40.140,0:06:43.560
Calls a calls B.

0:06:44.080,0:06:47.470
Minus. Sign a

0:06:47.470,0:06:53.658
sign be. Now this looks[br]very unwieldy and it would be

0:06:53.658,0:06:59.006
nice if in the same way that[br]this is in terms of signs and

0:06:59.006,0:07:03.972
causes and this ones in terms of[br]signs and causes. I could have

0:07:03.972,0:07:08.174
tan of A+B somehow in terms of[br]tangent and possibly cotangent,

0:07:08.174,0:07:12.758
but certainly I want to have[br]some tangents in there. So what

0:07:12.758,0:07:17.342
can I do? Well, look at this[br]term here cause a Cosby.

0:07:18.010,0:07:24.070
Supposing I divided everything[br]by that term cause a Cosby.

0:07:24.600,0:07:29.136
Well, that would give me one[br]here and I'd have this over

0:07:29.136,0:07:34.050
cause A cause be of course I'd[br]have sign over cause sign over

0:07:34.050,0:07:38.964
cause for each of A&B so I have[br]tan a tan be there.

0:07:39.990,0:07:47.153
Would I get anything nice on the[br]top? Well, let's write it down

0:07:47.153,0:07:54.485
in full. So we're[br]going to divide

0:07:54.485,0:07:57.908
everything by calls

0:07:57.908,0:08:05.115
a Cosby. And[br]I have to divide

0:08:05.115,0:08:08.975
everything by this because

0:08:08.975,0:08:15.130
I've got. A pair of[br]equal signs here a balance. So

0:08:15.130,0:08:21.130
what I do to one side I must do[br]the other. I must do everything

0:08:21.130,0:08:23.930
to all of the terms to preserve

0:08:23.930,0:08:30.884
the equality. Now that looks[br]absolutely awful. Absolutely

0:08:30.884,0:08:37.028
massive algebra. So[br]how can we

0:08:37.028,0:08:43.172
make it any[br]simpler? Well, let's

0:08:43.172,0:08:49.316
just go back[br]to the denominator

0:08:49.316,0:08:52.388
here. This bottom

0:08:52.388,0:08:59.609
term. Cause a over cause[br]a cancels down Cosby over. Cosby

0:08:59.609,0:09:02.975
cancels down. This is just one.

0:09:03.819,0:09:09.993
Sinai over 'cause I is Tanay and[br]sign be over. Cosby is tan be.

0:09:10.529,0:09:13.297
So this is just tan a tan B.

0:09:14.189,0:09:19.253
Have a look at this. Well,[br]there's a common factor on top

0:09:19.253,0:09:24.739
and bottom here of Cosby. I can[br]cancel that out. Leaves me with

0:09:24.739,0:09:27.693
sign a over cause a tan a.

0:09:28.309,0:09:32.039
Here the causes go out.

0:09:32.559,0:09:39.559
Sign be over. Cosby[br]leaves me with Tan

0:09:39.559,0:09:43.379
B so. What we end

0:09:43.379,0:09:50.409
up with. Turn of[br]A+B is 10A.

0:09:51.059,0:09:54.589
Plus 10B.

0:09:55.919,0:10:02.351
All over 1[br]- 10 a

0:10:02.351,0:10:03.423
10B.

0:10:04.529,0:10:10.001
Now we can do the same[br]again for tan.

0:10:10.649,0:10:17.864
All A-B and we've got two ways[br]of approaching it, first of all.

0:10:18.499,0:10:23.749
We could work through this again[br]X set with tan of A-B is the

0:10:23.749,0:10:26.749
sign of a minus B over the cause

0:10:26.749,0:10:30.139
of A-B. Or we could work through

0:10:30.139,0:10:36.029
it again. By making the same[br]replacement as we did before,

0:10:36.029,0:10:41.419
and rewriting this as the time[br]of A plus minus B.

0:10:43.319,0:10:50.865
Let's be assured that what it is[br]going to give us, which ever way

0:10:50.865,0:10:52.482
we do it.

0:10:56.599,0:11:02.975
Is going[br]to be

0:11:02.975,0:11:07.873
that. And So what we've got now[br]our our six.

0:11:08.659,0:11:09.809
Addition formula.

0:11:11.319,0:11:17.109
Sign of A+B sign of A-B cause of[br]A+B cause of A-B. These are the

0:11:17.109,0:11:21.741
ones you must know and learn[br]from those four you can derive

0:11:21.741,0:11:27.145
these. It's a help if you know[br]them. If you can learn them. The

0:11:27.145,0:11:32.163
important thing is to be able to[br]recognize them when you see them

0:11:32.163,0:11:34.479
and recognize when you need to

0:11:34.479,0:11:40.289
use them. Let's have a look at[br]three fairly typical examples of

0:11:40.289,0:11:45.063
the use of these. In many ways,[br]it's practiced that we're

0:11:45.063,0:11:48.535
getting at recognizing these[br]particular formula, 'cause we

0:11:48.535,0:11:52.875
may need them more often when[br]we're doing other, more

0:11:52.875,0:11:57.299
complicated manipulations. So[br]first of all, let's have a look

0:11:57.299,0:11:58.619
at this particular problem.

0:11:59.129,0:12:02.813
If we know that sign of

0:12:02.813,0:12:05.729
A. Is 3/5.

0:12:07.039,0:12:08.519
And that the cause.

0:12:09.119,0:12:13.031
Of B is 5

0:12:13.031,0:12:20.151
thirteenths. Then[br]What's the sign

0:12:20.151,0:12:22.679
of A+B?

0:12:24.249,0:12:27.579
What's the cause

0:12:27.579,0:12:35.399
of A-B?[br]OK.

0:12:36.449,0:12:40.625
We've got sign a but we don't[br]know anything about cause a

0:12:40.625,0:12:43.668
seemingly. We've got Cosby.

0:12:44.579,0:12:47.351
And we really don't know[br]anything about sign be.

0:12:49.519,0:12:55.226
We don't know much about A and[br]be really because a could be

0:12:55.226,0:13:00.933
either an obtuse angle or an[br]acute angle. So we really need a

0:13:00.933,0:13:05.762
little bit more information. So[br]let's say that A&B are acute.

0:13:06.289,0:13:09.810
In alerts that both less than 90

0:13:09.810,0:13:14.267
degrees. Well, if the boat less[br]than 90 degrees, one of the

0:13:14.267,0:13:16.055
things we can do is represent

0:13:16.055,0:13:21.277
the angle. And the sign with a[br]right angle triangle. So I just

0:13:21.277,0:13:23.117
have a look at that.

0:13:24.479,0:13:27.509
Right angle triangle.

0:13:27.509,0:13:34.457
Right angle there the angle a[br]sign a is 3/5 opposite over

0:13:34.457,0:13:40.826
hypotenuse, so that's three. The[br]side opposite the angle A and

0:13:40.826,0:13:46.616
that's five the hypotenuse which[br]is always opposite the right

0:13:46.616,0:13:49.511
angle, always the longest side.

0:13:50.259,0:13:55.506
Pythagoras tells us that this[br]other side has to be 4.

0:13:58.199,0:14:01.961
And so now that we know the[br]adjacent side, we know

0:14:01.961,0:14:05.723
everything about angle a. We[br]know it sign, we know it's

0:14:05.723,0:14:09.143
cosine and if we want we can[br]find its tangent.

0:14:10.679,0:14:11.959
Let's do the same.

0:14:12.479,0:14:13.778
The angle be.

0:14:15.669,0:14:22.173
Here is the angle be in its[br]right angle triangle and were

0:14:22.173,0:14:25.967
told that Cosby is 5 over 13.

0:14:26.629,0:14:31.822
Cosine is adjacent over[br]hypotenuse. This is the side

0:14:31.822,0:14:37.015
that's adjacent, so there's[br]five. This is the hypotenuse,

0:14:37.015,0:14:42.785
the longest side, the side[br]opposite the right angle, so

0:14:42.785,0:14:49.605
that's 30. So using Pythagoras[br]theorem, this side is 12.

0:14:50.369,0:14:54.729
Now notice I said using[br]Pythagoras Theorem, but it was

0:14:54.729,0:14:59.961
as though I knew these three[br]455-1213. I do know them and

0:14:59.961,0:15:04.757
you've got to get to know them[br]as well. Pythagorean triples,

0:15:04.757,0:15:06.501
simple triples of integers,

0:15:06.501,0:15:11.371
whole numbers. That a Bay,[br]Pythagoras's theorem if you have

0:15:11.371,0:15:16.579
them at your fingertips, can we[br]call them easy? Easily? You find

0:15:16.579,0:15:20.919
this so much better in working[br]in trigonometry 'cause these

0:15:20.919,0:15:26.561
numbers are used an awful lot.[br]OK, then, let's have a look at

0:15:26.561,0:15:32.203
what we can do. Sign of A+B? The[br]addition formula tells us they

0:15:32.203,0:15:35.459
sign a. Cause B.

0:15:36.049,0:15:39.681
Close calls a sign

0:15:39.681,0:15:47.259
be. Equals.[br]Sign a we know is 3/5.

0:15:47.959,0:15:55.119
Times, Cosby, and we know[br]that one. It's five thirteenths

0:15:55.119,0:16:02.389
Plus cause a we can[br]read cause a off here

0:16:02.389,0:16:08.932
it's adjacent over hypotenuse so[br]it's 4 over 5.

0:16:08.939,0:16:16.019
Times by sign B and we[br]can read, sign be off this

0:16:16.019,0:16:20.739
triangle it's opposite over[br]hypotenuse 12 over 13.

0:16:21.619,0:16:29.199
Do the arithmetic 3 fives[br]are 15 and five 1365,

0:16:29.199,0:16:36.021
so that's 15 over 65[br]+ 4 twelve 48.

0:16:36.799,0:16:44.047
And 5:13's again are 65 and[br]so that gives me a fraction

0:16:44.047,0:16:45.859
63 over 65.

0:16:46.859,0:16:50.099
We had cause of

0:16:50.099,0:16:56.219
A-B. So let's approach that in[br]exactly the same way.

0:16:56.919,0:17:00.439
Will quote our[br]expansion. Our

0:17:00.439,0:17:04.663
addition formula cause[br]of a Cosby.

0:17:05.759,0:17:09.244
Plus sign of a sign

0:17:09.244,0:17:16.957
of B. Substitute our[br]values cause a That's four

0:17:16.957,0:17:20.877
over 5 adjacent over hypotenuse.

0:17:22.189,0:17:28.909
Times, Cosby. That's five[br]over 13, adjacent over

0:17:28.909,0:17:29.749
hypotenuse.

0:17:30.789,0:17:37.869
Plus sign a That's opposite[br]over hypotenuse, three over 5.

0:17:39.009,0:17:45.457
Times by sign B.[br]That's 12 over 13,

0:17:45.457,0:17:47.875
opposite over hypotenuse.

0:17:49.039,0:17:53.549
Now here we've got some[br]fractions and it might be

0:17:53.549,0:17:58.510
tempting to cancel the fives[br]here. Five goals, five goes, but

0:17:58.510,0:18:05.726
five times by 13 is the 65 and[br]five times by 13 is the 65 to

0:18:05.726,0:18:07.530
have the same denominator.

0:18:08.319,0:18:15.951
So I'm not going to cancel.[br]4 fives are 20 over five

0:18:15.951,0:18:18.229
1365. Plus

0:18:19.499,0:18:24.317
Three twelves[br]are 36 over that

0:18:24.317,0:18:29.938
65 again giving[br]me 56 over 65.

0:18:31.049,0:18:35.479
So that's one way in which these[br]can be used.

0:18:35.529,0:18:37.881
Let's have a look at another

0:18:37.881,0:18:44.156
way. Supposing we were asked to[br]find what's sign 75, but no

0:18:44.156,0:18:49.760
don't reach for your Calculator,[br]you've got to workout sign 75 as

0:18:49.760,0:18:55.364
an expression, not as a set of[br]decimals, but as an expression.

0:18:55.364,0:19:01.902
What we've got to think about is[br]how can we make 75 from angles

0:19:01.902,0:19:04.704
who sign and cosine's? We know

0:19:04.704,0:19:12.133
very well. Well, 45[br]+ 30 gives us

0:19:12.133,0:19:19.699
75. I'm 45 and 30[br]are two of the angles that

0:19:19.699,0:19:26.339
we know sines and cosines for[br]exact sines and cosines.

0:19:26.869,0:19:34.309
So this would be sign[br]of 45 cause of 30.

0:19:35.259,0:19:42.819
Close calls of[br]45. Sign of

0:19:42.819,0:19:50.543
30. Sign 45[br]that's one over Route

0:19:50.543,0:19:57.619
2. Times by cost 30[br]will cost 30 is Route 3

0:19:57.619,0:19:58.707
over 2.

0:19:59.859,0:20:05.241
Plus cause of 45 is one[br]over Route 2.

0:20:05.819,0:20:11.779
Times by sign of 30, which[br]is just a half.

0:20:12.519,0:20:19.239
So we have Route 3. That's one[br]times by Route 3 over 2 Route 2.

0:20:19.859,0:20:26.827
Plus one over and again two[br]times route 2 is 2 Route 2.

0:20:26.827,0:20:31.651
We've got the same denominator,[br]so 2 Route 2.

0:20:32.229,0:20:39.335
To Route 2 and on the[br]top Route 3 + 1.

0:20:39.919,0:20:44.315
Now leave not won't like that we[br]might try and get rid of this

0:20:44.315,0:20:48.083
Route 2 in the denominator, but[br]just for the moment, let's leave

0:20:48.083,0:20:52.165
it like that. There's nothing to[br]be gained by doing it at this

0:20:52.165,0:20:53.735
stage, and let's take another

0:20:53.735,0:20:56.879
example. Let's have a look.

0:20:56.879,0:21:03.539
An 50 What we[br]need to be able to do is to find

0:21:03.539,0:21:07.379
15 in terms of angles that we[br]know lots of things about.

0:21:09.129,0:21:11.157
And of course.

0:21:11.679,0:21:18.271
One of the ways of doing this,[br]but only one is to do 60 - 45.

0:21:19.609,0:21:26.089
So that means that we need the[br]addition formula that's to do

0:21:26.089,0:21:33.109
with tangent and the one that's[br]to do with the tan of A-B.

0:21:33.109,0:21:36.349
So this is 1060 - 1045.

0:21:37.099,0:21:43.823
Over 1[br]+ 10

0:21:43.823,0:21:51.043
sixty 10:45.[br]So now let's put the numerical

0:21:51.043,0:21:58.099
values in. Now the tangent of 60[br]is Route 3 and the tangent of

0:21:58.099,0:21:59.611
45 is one.

0:22:00.399,0:22:06.187
Over. One plus[br]Route 3.

0:22:07.349,0:22:09.569
Times by one.

0:22:10.119,0:22:17.707
So we'll have root 3 - 1[br]over Route 3 times by one is

0:22:17.707,0:22:19.333
just Route 3.

0:22:20.469,0:22:26.919
Plus one. Now that's[br]all right, and it's correct.

0:22:27.769,0:22:32.241
Doesn't look very nice and we[br]tend to have a tradition of not

0:22:32.241,0:22:35.681
leaving things like root 3 plus[br]one in the denominator.

0:22:36.239,0:22:39.753
So one of the ways of tidying

0:22:39.753,0:22:46.859
that up. Is to multiply top[br]and bottom of this fraction by

0:22:46.859,0:22:48.413
the same thing?

0:22:49.249,0:22:54.829
Multiplying top and bottom by[br]the same thing keeps it the same

0:22:54.829,0:22:59.944
value, but what should we[br]multiply it by? Well, I'm going

0:22:59.944,0:23:03.199
to choose to multiply by Route 3

0:23:03.199,0:23:10.029
- 1. Not be cause that's what's[br]in the numerator there, but

0:23:10.029,0:23:16.519
because this is the difference[br]of two squares A+B, A-B and

0:23:16.519,0:23:23.009
A+B times by A-B, let's just[br]write that down up here.

0:23:23.039,0:23:26.779
Is A squared minus B

0:23:26.779,0:23:33.990
squared? So that means that[br]on the bottom I have Route 3

0:23:33.990,0:23:36.545
times by Route 3A squared.

0:23:37.609,0:23:43.057
Which is just three minus B[br]squared. B was just one, so

0:23:43.057,0:23:49.867
that's just minus one and 3 - 1[br]is 2 and integer not absurd. Not

0:23:49.867,0:23:55.315
one of these things with a[br]square root sign attached to it.

0:23:55.315,0:24:00.763
What have we got on top? We two[br]brackets that we're multiplying

0:24:00.763,0:24:07.573
together route 3 by Route 3 is[br]3, route 3 by minus one is minus

0:24:07.573,0:24:14.794
Route 3. 1 by minus one[br]by Route 3 is minus Route 3

0:24:14.794,0:24:18.469
and minus one with minus one is

0:24:18.469,0:24:24.809
plus one. 3 plus One is 4 -[br]2 RT. Threes were taking away

0:24:24.809,0:24:30.969
all over 2 and now there's a[br]common factor of 2 on the top

0:24:30.969,0:24:38.009
and on the bottom 2 into 4 goes[br]to an two into minus 2. Route 3

0:24:38.009,0:24:39.769
goes minus Route 3.

0:24:40.339,0:24:44.785
And two into two goes wall, so[br]we needn't worry about that one.

0:24:44.785,0:24:49.231
So there we've got a second[br]example to add to the first one

0:24:49.231,0:24:51.625
that we did. Let's take one more

0:24:51.625,0:24:57.869
type. Of example, where we can[br]make you solve the addition

0:24:57.869,0:25:04.313
formula and this is where we get[br]to simplify an expression. So

0:25:04.313,0:25:09.146
supposing we've got the sign of[br]90 plus A.

0:25:10.069,0:25:15.505
Could we have this simpler?[br]Could we just have, say, sign a

0:25:15.505,0:25:21.394
or minus sign a or cause a or[br]something like that? Does it

0:25:21.394,0:25:27.736
have to be 90 plus a? Well,[br]let's have a look. It's sign of

0:25:27.736,0:25:31.069
90. Calls of

0:25:31.069,0:25:37.819
a. Plus the[br]cause of 90

0:25:37.819,0:25:40.729
sign of A.

0:25:42.009,0:25:44.357
Sign 90 is one.

0:25:44.979,0:25:50.724
Cause I just cause I so first of[br]all we have one times cause a

0:25:50.724,0:25:52.639
which is just cause a.

0:25:53.759,0:25:55.967
But cause of 90.

0:25:56.499,0:26:02.969
Is 0. Times by sign a[br]well, anything times by zero is

0:26:02.969,0:26:06.641
0 so we just left with cause a.

0:26:07.599,0:26:15.027
So an expression such a sign[br]of 90 plus a reducers tool

0:26:15.027,0:26:21.836
cause a what about something[br]like the cosine of 180 minus

0:26:21.836,0:26:28.729
a? Well, this[br]is cause 180

0:26:28.729,0:26:34.383
cause a.[br]Plus sign

0:26:34.383,0:26:37.307
180 sign

0:26:37.307,0:26:44.421
a. The cosine of[br]180. That's minus one, so minus

0:26:44.421,0:26:52.093
one times by cause a is minus[br]cause a sign of 180 zero and

0:26:52.093,0:26:58.669
sign is just sign a but zero[br]times by anything is 0.

0:26:59.279,0:27:05.817
So we're just left with minus[br]calls a so we can see that these

0:27:05.817,0:27:10.954
addition formula help us to very[br]quickly simplify and get in

0:27:10.954,0:27:13.289
terms of this angle a.

0:27:14.109,0:27:16.479
Possibly quite complicated

0:27:16.479,0:27:23.355
expressions. But the basic thing[br]is that these 6 addition

0:27:23.355,0:27:29.431
formula. Again, I stress it. You[br]have to learn you have to know.

0:27:29.431,0:27:33.241
But most importantly, you've got[br]to recognize them and recognize

0:27:33.241,0:27:37.813
their use when you see them and[br]when you see the situation.