WEBVTT

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Hi, I'm Fiona.

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Let's have a look at TAMUA 2021 paper 2
and question 19.

69d5506b-cdb7-4402-831e-bc5e51aab2e6-0
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We're told that the angle Theta can take
any of the values 123 and so on up to

69d5506b-cdb7-4402-831e-bc5e51aab2e6-1
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360°,
and we're asked for how many of these

69d5506b-cdb7-4402-831e-bc5e51aab2e6-2
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values of Theta is it true that sine
Theta times sqrt 1 plus sine Theta times

69d5506b-cdb7-4402-831e-bc5e51aab2e6-3
00:00:29.286 --> 00:00:36.007
sqrt 1 minus sine Theta plus Cos Theta
times sqrt 1 plus Cos Theta times sqrt 1

69d5506b-cdb7-4402-831e-bc5e51aab2e6-4
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minus Cos Theta equals zero?

21c18661-6eee-4340-a84c-7b6787a3dd94-0
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And then we're given options A to H to
choose from once we get to that point.

dd89ecb8-c61c-4347-9477-ee7441e92d16-0
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Now,
when I look at the equation that I'm

dd89ecb8-c61c-4347-9477-ee7441e92d16-1
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given,
I can see sine and I can see cosine.

2dbb6737-77a2-41a6-a066-57301ad060e0-0
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I know that I'm also considering values
of Theta between 1 and 360 degrees.

465fcf52-592b-4595-b248-c19144624f69-0
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So I'm going to start by sketching a
graph of both sine and cosine Theta

465fcf52-592b-4595-b248-c19144624f69-1
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between those values of Theta and I'm
going to overlay them on top of one

465fcf52-592b-4595-b248-c19144624f69-2
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another.

48cc61dc-147e-4af2-91c8-f5ad077c7dba-0
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So here's my graph.

9ab30f87-690d-452d-86fe-1272c930bb02-0
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Square sine Theta Y equals sine Theta is
in yellow and Y equals Cos Theta is in

9ab30f87-690d-452d-86fe-1272c930bb02-1
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orange.

539da477-9837-4432-8732-2e4b7d8e0896-0
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Now back to this equation.

459ab9e0-6636-4764-a293-50032835a6ea-0
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I can see here that I've got sqrt 1 plus
sine Theta and sqrt 1 minus sine Theta,

459ab9e0-6636-4764-a293-50032835a6ea-1
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and I can put that together under one
square root.

36113b29-515e-4b29-acce-ab7af0b191b9-0
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So let's do that.

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I can also do the same for the term
involving cosine.

f47e505d-7414-4c22-870a-a32417f6e6ba-0
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OK, now looking at these terms,
I can see that one plus sine Theta times

f47e505d-7414-4c22-870a-a32417f6e6ba-1
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1 minus sine Theta is a factorization of
the difference of two squares.

f3e553b5-6a18-4e8c-84eb-60c6a87e3693-0
00:02:01.880 --> 00:02:08.493
So this term here is equivalent to sine
Theta times sqrt 1 ^2,

f3e553b5-6a18-4e8c-84eb-60c6a87e3693-1
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which is 1 minus sine squared Theta,
and we'll notice a similar pattern for

f3e553b5-6a18-4e8c-84eb-60c6a87e3693-2
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the term involving cosine.

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Next,
I notice that I have one minus sine

208d6d28-150c-492b-88ff-e559d0138ee3-1
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squared Theta and 1 minus Cos squared
Theta,

208d6d28-150c-492b-88ff-e559d0138ee3-2
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and I'm thinking about that identity that
sine squared Theta plus Cos squared Theta

208d6d28-150c-492b-88ff-e559d0138ee3-3
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equals 1.

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So that means that one minus sine squared
Theta is equivalent to Cos squared Theta.

9edb6496-8be4-4f53-8ecc-70ac6a4d92f2-0
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So this first term is equivalent to sine
Theta times the square root of Cos

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squared Theta.

d60dd95c-ebc2-43ed-8321-2dd7a31544df-0
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And this second term is equivalent to Cos
Theta times the square root of sine

d60dd95c-ebc2-43ed-8321-2dd7a31544df-1
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squared Theta.

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And this all equals 0.

9141009d-06a9-495f-b1f4-bcac3a79951a-0
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Now it might be tempting at this point to
have a look at this square root of Cos

9141009d-06a9-495f-b1f4-bcac3a79951a-1
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squared Theta and think, well,
that's just Cos Theta or this square root

9141009d-06a9-495f-b1f4-bcac3a79951a-2
00:03:07.561 --> 00:03:11.440
of sine squared Theta and think, well,
that's just sine Theta.

4071880f-2732-4fda-aec7-72adfbf0e994-0
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But we need to take a moment to reflect
here because this square root can only

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output positive values.

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Let's take a numerical case.

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If we were to have sqrt -3 ^2,
then if we were to evaluate this,

2f3fd9b2-8d36-4cac-b4ba-9f725e246016-1
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this would have to take a value of 3.

c5c4df5a-0da6-4856-abfe-49f6bc4ffd95-0
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And so if we were to think about this as
being the same as the modulus of -3,

c5c4df5a-0da6-4856-abfe-49f6bc4ffd95-1
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that would give us the output that we
want.

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So now looking back at our equation that
we're dealing with,

f60adfa3-8197-4179-99d5-53847a811f49-1
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this equation is in fact equivalent to
sin Theta times the modulus of Cos Theta

f60adfa3-8197-4179-99d5-53847a811f49-2
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plus Cos Theta times the modulus of times
the modulus of sine Theta,

f60adfa3-8197-4179-99d5-53847a811f49-3
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and that all equals 0.

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Now I'm just going to take a moment to
tidy up the board here because what we've

bb48dd58-f732-45f5-b4e8-fb0421835f6d-1
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discovered is that this equation that we
started with is equivalent to this one

bb48dd58-f732-45f5-b4e8-fb0421835f6d-2
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that we've ended up with here.

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So I'm just going to kind of clear the
board off to make it give myself a bit

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more space.

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So here I am.

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I've arrived at sine Theta times modulus
of Cos Theta plus Cos Theta times the

3c44954c-4e04-4e82-83d1-fd9b3ea7f9a4-1
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modulus of sine Theta equals 0,
and that tells me that I'm going to need

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to consider the different cases for which
cosine Theta and sine Theta are greater

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than or equal to 0 or are less than 0.

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Let me write a table up so that I can
keep track of the cases that I want to

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consider.

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OK,
now I can see that I'm going to need to

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consider four different cases for the
values of sine Theta and cosine Theta.

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Let's start with case one where sine
Theta and Cos Theta are both greater than

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or equal to zero.

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Well, if that is the case,
then this equation here,

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let's label this with case one would be
sine Theta, Cos Theta plus Cos Theta.

31787f19-d8bd-410f-9d36-8272f893f313-0
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Sine Theta equals 0,
so I'm going to say that that's the same

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as sine Theta Cos Theta plus sine Theta.

c75b5e82-3627-4aeb-a991-e51d27177262-0
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Cos Theta equals 0.

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So that would give me two sine Theta Cos
Theta equals 0.

a0d74f8a-2761-4f8a-8668-fbd149f0333e-0
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Then if I divide all across by two,
I would end up with sine Theta Cos Theta

a0d74f8a-2761-4f8a-8668-fbd149f0333e-1
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equals 0.

5a3d08f8-7006-41c3-8f50-e80d4f2e15b8-0
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Now this can only hold when sine Theta or
Cos Theta takes a value of 0.

a38e1c33-8f13-4586-9b1f-7f94d2111f7c-0
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And this is where my graph will come in
handy.

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Let me grab another piece of coloured
chalk.

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So let's look at our range of values of
Theta.

f4bc0961-409b-4e28-a59d-0242b69a13db-0
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Now let's not forget that Theta cannot
take a value of 0°.

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So I'm not going to be looking at any
solutions here.

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But as I come along,
the first time that one of my functions

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equals 0 is at Theta equals 90°,
and that's when Cos Theta equals 0.

15d34f8f-6509-4e60-bd85-0db53d090047-0
00:06:23.280 --> 00:06:30.525
Then when Theta is 180°,
sine Theta equals 0, when Theta is 270°,

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Cos Theta equals zero,
and then when Theta is 360°,

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Cos Theta equals 0.

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So I'm marking all of the possible values
of Theta that I'm finding as I go along.

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And for case one,
I found four possible values of Theta.

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Now let's jump ahead to case 4 because
case 4 looks at when both of these are

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negative or a less than 0.

02b6a08e-624c-43bb-856f-664f9776ab3a-0
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So when this is the case,
we would end up with sine Theta minus Cos

02b6a08e-624c-43bb-856f-664f9776ab3a-1
00:07:05.714 --> 00:07:10.200
Theta plus Cos Theta minus sine Theta
equals 0.

88e705f4-95ca-4120-8779-1ca15d4a3e39-0
00:07:10.520 --> 00:07:15.858
So for the modulus function,
let's say if we're dealing with modulus

88e705f4-95ca-4120-8779-1ca15d4a3e39-1
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of X, if X is negative,
then the modulus function needs to apply

88e705f4-95ca-4120-8779-1ca15d4a3e39-2
00:07:20.886 --> 00:07:24.600
minus X in order to get back a positive
output.

fcd4d9ad-2aaf-49c1-bc14-f6f2f3b9705c-0
00:07:25.280 --> 00:07:31.295
So in for case four,
we would end up with -2 sine Theta Cos

fcd4d9ad-2aaf-49c1-bc14-f6f2f3b9705c-1
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Cos Theta equals 0.

f911ad29-4f01-4cb7-b03a-80bd2d4c4620-0
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And you can see that if I divide all
across by -2, then case four,

f911ad29-4f01-4cb7-b03a-80bd2d4c4620-1
00:07:38.032 --> 00:07:43.400
let me label this as case 4 would bring
me to the same conclusion as case one.

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Now let's move on to case 2 that I want
to consider.

07b7429a-49f3-4212-b15d-5956dbe0346f-0
00:07:48.480 --> 00:07:55.002
Case 2 is when sine Theta is greater than
or equal to 0 and Cos Theta is less than

07b7429a-49f3-4212-b15d-5956dbe0346f-1
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0.

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OK,
well when sine Theta is greater than or

267c4121-5ff9-4d53-afc7-17d86ec3dfe2-1
00:07:58.756 --> 00:08:01.952
equal to 0,
then the modulus of sine Theta is just

267c4121-5ff9-4d53-afc7-17d86ec3dfe2-2
00:08:01.952 --> 00:08:03.080
sine Theta itself.

a3bb2c22-2c92-4748-9d58-6b0dd66ab7d8-0
00:08:03.520 --> 00:08:08.675
When Cos Theta is less than 0,
the modulus of Cos Theta is going to be

a3bb2c22-2c92-4748-9d58-6b0dd66ab7d8-1
00:08:08.675 --> 00:08:12.960
minus Cos Theta in order to get back that
positive output.

5ff09b5e-9f56-47f3-ba68-864f9976cb66-0
00:08:13.360 --> 00:08:18.160
So we would end up with minus sine Theta,
Cos Theta plus Cos Theta sine Theta.

7605199b-72d8-4dba-8374-ff0e7c0d3c7f-0
00:08:18.160 --> 00:08:19.360
Let's write that down.

f05a8cce-c937-4134-9e51-9a962fb3405b-0
00:08:23.560 --> 00:08:27.288
Now,
just with a little bit of rearranging,

f05a8cce-c937-4134-9e51-9a962fb3405b-1
00:08:27.288 --> 00:08:34.067
I can say that this is exactly the same
as Cos Theta sine Theta minus Cos Theta

f05a8cce-c937-4134-9e51-9a962fb3405b-2
00:08:34.067 --> 00:08:35.000
sine Theta.

03610c00-db1d-461f-908f-1f2ee03cd1d9-0
00:08:35.280 --> 00:08:39.292
Well,
this is going to equal 0 for all values

03610c00-db1d-461f-908f-1f2ee03cd1d9-1
00:08:39.292 --> 00:08:42.520
of Theta which satisfy this equation.

cfc3fd4c-7be7-4937-8457-da19ef3a7d3b-0
00:08:42.760 --> 00:08:47.414
And we know that that is when sine Theta
is greater than or equal to 0 and at the

cfc3fd4c-7be7-4937-8457-da19ef3a7d3b-1
00:08:47.414 --> 00:08:50.480
same time as Cos Theta being less than or
equal to 0.

6ab2b3d7-b5f4-4f2b-8300-959ffde25a60-0
00:08:51.280 --> 00:08:56.000
Let's now also consider case 3 and how it
relates to case 2.

21d3d7dc-c307-442f-b5f2-1f690a824f02-0
00:08:56.320 --> 00:08:59.483
So in case 3,
we would have sine Theta being less than

21d3d7dc-c307-442f-b5f2-1f690a824f02-1
00:08:59.483 --> 00:09:02.360
0 and Cos Theta being greater than or
equal to 0.

2eea930e-c066-4605-97a2-96c798294dad-0
00:09:02.720 --> 00:09:05.440
We would end up with the same result.

c9b17e7c-e493-4c26-ad81-db7a94e00a34-0
00:09:05.680 --> 00:09:08.840
Cos Theta sine Theta minus Cos Theta sine
Theta equals zero.

6871508c-2739-4dff-80d2-e963e7c21397-0
00:09:08.840 --> 00:09:14.413
And of course Cos Theta sine Theta minus
Cos Theta sine Theta is going to equal 0

6871508c-2739-4dff-80d2-e963e7c21397-1
00:09:14.413 --> 00:09:17.880
for all values of Theta for which this
case holds.

aa92ec9c-6757-4c58-b09d-82a0061ab3a2-0
00:09:17.880 --> 00:09:21.800
And that is when sine Theta is negative
and Cos Theta is positive.

8ad65c1a-b66a-4a32-9735-606e64244682-0
00:09:22.240 --> 00:09:24.120
Now how does this relate graphically?

cc5cf354-1fe0-4892-917b-4d0292c163d1-0
00:09:24.120 --> 00:09:30.858
And this is where the power of drawing my
sketch comes in because I want to look

cc5cf354-1fe0-4892-917b-4d0292c163d1-1
00:09:30.858 --> 00:09:37.263
for in case 2 and in case sorry and in
case 3 I want to look for areas on my

cc5cf354-1fe0-4892-917b-4d0292c163d1-2
00:09:37.263 --> 00:09:43.918
graph or values of Theta for which the
graph of these functions satisfies these

cc5cf354-1fe0-4892-917b-4d0292c163d1-3
00:09:43.918 --> 00:09:45.000
two criteria.

1c2521ed-3e9f-4bcc-b2aa-40544b793054-0
00:09:45.160 --> 00:09:49.295
So the first one is when sine Theta is
positive well greater than or equal to 0

1c2521ed-3e9f-4bcc-b2aa-40544b793054-1
00:09:49.295 --> 00:09:50.640
and Cos Theta is negative.

fb154dd8-96bf-43a7-8fc5-f484b0a1594d-0
00:09:51.640 --> 00:09:57.800
So when sine Theta lies above the X axis
and Cos Theta lies below the X axis.

c811c61e-a7a2-4fba-9371-627d6f67b478-0
00:09:58.040 --> 00:10:02.458
And looking at my graph,
I can see that that is for values of

c811c61e-a7a2-4fba-9371-627d6f67b478-1
00:10:02.458 --> 00:10:04.240
Theta between 90 and 180.

b8e23942-54e9-4f8d-bed1-bc44001ace72-0
00:10:05.280 --> 00:10:10.414
Now case 3 is when sine Theta is negative
and Cos Theta is greater than or equal to

b8e23942-54e9-4f8d-bed1-bc44001ace72-1
00:10:10.414 --> 00:10:10.720
zero.

30a103a3-f88b-4ee8-91f9-7be0f2471d83-0
00:10:11.200 --> 00:10:17.135
Well here is where Cos Theta lies above
the X axis at the same time as sine Theta

30a103a3-f88b-4ee8-91f9-7be0f2471d83-1
00:10:17.135 --> 00:10:18.800
lying below the X axis.

0c3cdcd6-ea91-49d6-ab6b-57c4f6a785b5-0
00:10:19.080 --> 00:10:25.840
So when Theta lies between 270 and 360,
that's when I will satisfy case 3.

f0327665-ebbc-4207-8775-d531b05e8d93-0
00:10:26.040 --> 00:10:32.997
So now I have found the range of possible
values of Theta for which this original

f0327665-ebbc-4207-8775-d531b05e8d93-1
00:10:32.997 --> 00:10:34.440
equation is true.

a4d73edf-086e-493d-b204-b3fde780c1ce-0
00:10:34.920 --> 00:10:40.190
Now I need to go and count how many there
are because looking back at the the

a4d73edf-086e-493d-b204-b3fde780c1ce-1
00:10:40.190 --> 00:10:43.907
conditions on Theta or the values that
Theta can take,

a4d73edf-086e-493d-b204-b3fde780c1ce-2
00:10:43.907 --> 00:10:48.840
Theta can only take integer values 123,
etcetera, all the way up to 360.

fdbda2a4-be2b-4185-942d-c066bf6ad45f-0
00:10:49.160 --> 00:10:55.848
So now I'm looking at these two intervals
90 to 180 and 270 to 360 and I'm counting

fdbda2a4-be2b-4185-942d-c066bf6ad45f-1
00:10:55.848 --> 00:10:57.680
all the integer values.

a416eb23-844d-4049-9569-f9c8a0970dae-0
00:10:58.000 --> 00:11:00.808
Well,
how many integer values will be between

a416eb23-844d-4049-9569-f9c8a0970dae-1
00:11:00.808 --> 00:11:01.480
90 and 180?

8f9b7ce2-edb8-44d3-9fdb-90bd60b59756-0
00:11:01.720 --> 00:11:05.603
And don't forget we want to include the
value of 90,

8f9b7ce2-edb8-44d3-9fdb-90bd60b59756-1
00:11:05.603 --> 00:11:10.880
so the number of integer values that will
be in that range will be 180.

41524611-29ac-4d7e-9e85-c8a64cc407f1-0
00:11:10.960 --> 00:11:17.360
Take away 89 because I want to include
that value of 90.

fc27ff74-a220-40e4-b6a9-b76f622771d1-0
00:11:18.800 --> 00:11:24.567
I also want to count the number of
integer values in this interval and I

fc27ff74-a220-40e4-b6a9-b76f622771d1-1
00:11:24.567 --> 00:11:28.360
want to add that on and that will be 360
- 269.

cbe0a932-8e4d-44d5-817a-6cf76c1ebc60-0
00:11:32.640 --> 00:11:41.360
180 - 89 will be 91 and 360 - 269 will be
91.

755cd044-8a6f-4e4a-ad41-1e31f9082341-0
00:11:41.600 --> 00:11:48.984
So in total the number of values of Theta
for which this original equation is true

755cd044-8a6f-4e4a-ad41-1e31f9082341-1
00:11:48.984 --> 00:11:51.920
or holds is 91 + 91 which is 182.

aa544445-613e-495e-9234-58564180f9f2-0
00:11:52.280 --> 00:11:55.119
Now looking at the values I have to
choose from,

aa544445-613e-495e-9234-58564180f9f2-1
00:11:55.119 --> 00:11:58.480
I can see that option F is the value I'm
looking for 182.

9607b860-f119-465a-9af3-fe02c29a13f7-0
00:11:58.480 --> 00:12:01.000
So option F is the answer to this
question.