WEBVTT

0d31920a-01e5-4818-bacc-72989ddd3f13-0
00:00:05.560 --> 00:00:10.320
Hi I'm Fiona and let's have a look at
Tamila 2021 paper one.

9f52f7f1-6a4a-4ea8-9240-1c56c27fafdb-0
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In question 19,
we're given an equation sine squared of 4

9f52f7f1-6a4a-4ea8-9240-1c56c27fafdb-1
00:00:15.266 --> 00:00:19.360
to the power of cosine Theta times 60° =
3 / 4.

3eb58387-53dd-49f1-8ae5-672f7a4036e2-0
00:00:19.720 --> 00:00:25.683
We're told it has exactly 3 solutions
when Theta ranges in value from zero to X

3eb58387-53dd-49f1-8ae5-672f7a4036e2-1
00:00:25.683 --> 00:00:26.280
degrees.

7f7d974e-3131-49a5-8954-7d7687db8c26-0
00:00:26.640 --> 00:00:31.840
And we're asked what is the what is the
range of all possible values of X and

7f7d974e-3131-49a5-8954-7d7687db8c26-1
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given options A to F to choose from.

05d18ca1-add5-47a2-a071-efa9ab1e5824-0
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Now when I look at this equation,
I can see that it's quite complex.

0ae564cd-7d0c-4efc-b565-4e1add4dfd35-0
00:00:39.160 --> 00:00:41.120
There's it's got many parts to it.

e56a5e4c-2baf-489e-9564-e75939dee278-0
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So I'm going to start with what is
familiar to me.

5ffb5657-a351-453b-b271-31482fbb3d28-0
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I'm going to think about when sine
squared of something which I will

5ffb5657-a351-453b-b271-31482fbb3d28-1
00:00:50.968 --> 00:00:54.800
represent as a box with a box equals 3 /
4.

7cca0a7b-ee0c-46a5-a0cb-8e3cb5eedb1d-0
00:00:55.160 --> 00:00:59.544
Now if sine squared of a thing equals 3 /
4,

7cca0a7b-ee0c-46a5-a0cb-8e3cb5eedb1d-1
00:00:59.544 --> 00:01:06.657
that means that sine of that thing equals
plus or minus sqrt 3 / sqrt 4,

7cca0a7b-ee0c-46a5-a0cb-8e3cb5eedb1d-2
00:01:06.657 --> 00:01:14.257
which is 2 and And when I think about
sine of something equalling root 3 / 2,

7cca0a7b-ee0c-46a5-a0cb-8e3cb5eedb1d-3
00:01:14.257 --> 00:01:17.960
my reference angle is an angle of 60°.

039b1d0c-6997-498a-857b-32d0a4a5754c-0
00:01:18.280 --> 00:01:24.520
So I'm going to think about positive
inputs with a reference angle of 60°.

983fe3ce-e773-400b-9375-3f13b737e7bb-0
00:01:24.720 --> 00:01:29.340
The reason I'm thinking about positive
inputs is because the the box represents

983fe3ce-e773-400b-9375-3f13b737e7bb-1
00:01:29.340 --> 00:01:33.325
what's in the bracket up here,
and 4 to the power of cosine Theta is

983fe3ce-e773-400b-9375-3f13b737e7bb-2
00:01:33.325 --> 00:01:37.079
going to be positive even if cosine Theta
takes negative values.

f9f0c06d-5230-4ed6-bd50-c4fb822e38a0-0
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And 60° is a positive value.

10e78e64-91a7-47d1-994c-d407bb06f7e7-0
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So whatever's in that bracket will always
be positive.

aa6eab87-e5f1-45b6-b1c6-0c1710666d6b-0
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I know that.

8bbcb090-e5d3-4c98-960e-1d92ffc58d34-0
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So let's think about the graph of the
sine function for positive inputs.

1289b462-2888-4b43-95f8-b8e4beedaa6a-0
00:01:52.720 --> 00:01:57.752
OK, on this graph,
the value of root 3 / 2 would be around

1289b462-2888-4b43-95f8-b8e4beedaa6a-1
00:01:57.752 --> 00:02:02.017
about here,
and the value of minus root 3 / 2 two

1289b462-2888-4b43-95f8-b8e4beedaa6a-2
00:02:02.017 --> 00:02:04.320
would be around about here.

d59f1b53-34d4-485b-9fa0-82e24ea22d47-0
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And as I go along this dotted line here,
when I hit the graph,

d59f1b53-34d4-485b-9fa0-82e24ea22d47-1
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I'm hitting solutions to this equation.

613574f9-72b4-4767-9953-bfee01ff6048-0
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Here the first solution I would hit would
be 60°,

613574f9-72b4-4767-9953-bfee01ff6048-1
00:02:17.380 --> 00:02:21.160
second solution I hit would be 120° and
so on.

132f03e4-1e44-4eb9-b967-d3e0f3f1ea52-0
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I would also hit solutions along this
dotted line for minus root 3 / 2.

5011a738-2237-405d-9782-8dff055852e8-0
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So that means that this thing here
represented by a box can take a number of

5011a738-2237-405d-9782-8dff055852e8-1
00:02:34.326 --> 00:02:34.880
values.

44b75284-62bc-484a-8943-d6424f51625b-0
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The first one would be 60°,
that was my first solution I hit,

44b75284-62bc-484a-8943-d6424f51625b-1
00:02:39.891 --> 00:02:41.880
Second one would be 120°.

77b01f0f-4353-48ab-9fe5-e3ad81e29e18-0
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And then as I go along I would hit more
values.

8e3f3920-fa85-489a-91b4-a04de8718a87-0
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So let me write those in.

288969db-ca04-4c6d-90a6-f9b76ea8eea8-0
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OK,
I've chosen to take the first five values

288969db-ca04-4c6d-90a6-f9b76ea8eea8-1
00:02:54.130 --> 00:02:58.653
that this thing can take,
but of course there would be more,

288969db-ca04-4c6d-90a6-f9b76ea8eea8-2
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so I'll add a...

0733ae20-8dee-4d85-b490-f1d5a95b723e-0
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At this stage,
I'm just taking more values than I think

0733ae20-8dee-4d85-b490-f1d5a95b723e-1
00:03:03.710 --> 00:03:05.880
I will need to be on the safe side.

90b64fac-418f-4297-af69-dfd49a0e2862-0
00:03:06.360 --> 00:03:08.400
So let's come back to this box.

2d96865c-cccb-47f1-8c9d-def73ee45dd2-0
00:03:08.400 --> 00:03:08.520
Now.

ed2d757b-6cae-453e-b46b-516e1c7d08a9-0
00:03:08.520 --> 00:03:11.120
This box represents what's in the bracket
up here.

38dae591-41bc-4f1c-91e1-ef3a9ae440f8-0
00:03:11.120 --> 00:03:18.400
So let's write that in and four to the
power of cosine Theta times 60°.

03b98532-a01e-40d8-ab9a-f015011b4122-0
00:03:18.680 --> 00:03:25.147
Now if 4 to the power of cosine Theta
times 60° can take these values,

03b98532-a01e-40d8-ab9a-f015011b4122-1
00:03:25.147 --> 00:03:32.251
then that means 4 to the power of cosine
Theta can take the what I get when I

03b98532-a01e-40d8-ab9a-f015011b4122-2
00:03:32.251 --> 00:03:35.440
divide each of these values by 60°.

9849bea7-2078-42c4-9985-66d5c4473c89-0
00:03:35.920 --> 00:03:45.157
So 60 / 60 is 120 / 60 is 2,
240 / 60 is four,

9849bea7-2078-42c4-9985-66d5c4473c89-1
00:03:45.157 --> 00:03:51.840
300 / 60 is 5 and so on and so on.

a22b0fac-75e5-4807-9e5c-ce3dcaea50eb-0
00:03:52.960 --> 00:03:58.864
Now if 4 to the power of cosine Theta can
take these values,

a22b0fac-75e5-4807-9e5c-ce3dcaea50eb-1
00:03:58.864 --> 00:04:06.123
then I want to see what values cosine
Theta can take and I will find those

a22b0fac-75e5-4807-9e5c-ce3dcaea50eb-2
00:04:06.123 --> 00:04:11.640
values by thinking about or doing the log
to the base 4.

6c4f1984-4eba-4ec8-bc08-06a71825b344-0
00:04:11.960 --> 00:04:16.074
So if I take log to the base four of four
to the power of cosine Theta,

6c4f1984-4eba-4ec8-bc08-06a71825b344-1
00:04:16.074 --> 00:04:17.160
I get cosine Theta.

c681e8c6-6f2f-4cc8-b426-e4edd98ca0b1-0
00:04:17.680 --> 00:04:25.087
And now if I take log to the base four of
one, well 4 to the power of 0 = 1,

c681e8c6-6f2f-4cc8-b426-e4edd98ca0b1-1
00:04:25.087 --> 00:04:32.591
so I get a value of 0 and log to the base
four of two would be a value of 1/2

c681e8c6-6f2f-4cc8-b426-e4edd98ca0b1-2
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because 4 to the power of 1/2 or sqrt 4 =
2.

ff6a8292-2bae-4f32-800c-195ee05909f7-0
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Log to the base four of four would give
me a value of 1 because 4 to the power of

ff6a8292-2bae-4f32-800c-195ee05909f7-1
00:04:45.179 --> 00:04:45.840
1 is 4.

8fb75121-e5d1-441e-9bf9-53472163439f-0
00:04:47.160 --> 00:04:53.720
Now if I think about log to the base four
of five, well, 5 is greater than four.

29731174-d206-4493-b3c4-880b68b27ec4-0
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That means that the power I would need to
put on 4 in order to get 5 would be

29731174-d206-4493-b3c4-880b68b27ec4-1
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greater than one.

13ff607f-c7ca-4781-b6e9-127bf49dae98-0
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But of course,
the cosine Theta can only take a maximum

13ff607f-c7ca-4781-b6e9-127bf49dae98-1
00:05:02.802 --> 00:05:03.400
value of 1.

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So I'm not going to add a...

a7ab778c-97c3-4a3a-98b5-a4222ec35107-0
00:05:05.320 --> 00:05:05.600
here.

e08fe297-f0a8-4d6f-b839-1b62ee8d53d2-0
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These are the three solutions that I can
take.

3d290875-1f15-4721-9f38-0219fbc5f68d-0
00:05:08.880 --> 00:05:12.741
They're not the three solutions that I'm
looking for here,

3d290875-1f15-4721-9f38-0219fbc5f68d-1
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but they are the only three solutions
that cosine Theta can take or the only

3d290875-1f15-4721-9f38-0219fbc5f68d-2
00:05:17.781 --> 00:05:20.400
three values that cosine Theta can take.

bdef35f0-57be-4921-a8e2-f8cff84dc2c8-0
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Right now we need to think about the the
graph of cosine Theta for positive values

bdef35f0-57be-4921-a8e2-f8cff84dc2c8-1
00:05:27.681 --> 00:05:32.080
because we know Theta will only take
positive values.

61e16c1b-9e4c-4df4-85ab-70e08f5bffb9-0
00:05:32.080 --> 00:05:36.436
So let's consider cosine Theta for
positive values and see where we get

61e16c1b-9e4c-4df4-85ab-70e08f5bffb9-1
00:05:36.436 --> 00:05:36.800
there.

1f62c3af-3480-4ed5-9471-109f68f50aa7-0
00:05:44.640 --> 00:05:49.120
So here's my graph of cosine Theta for
positive values of Theta.

1943e8b5-358f-400a-af68-551ea2796258-0
00:05:49.760 --> 00:05:52.466
Now,
what does it mean for cosine Theta to

1943e8b5-358f-400a-af68-551ea2796258-1
00:05:52.466 --> 00:05:53.600
take these values?

aee8d6a6-b94a-411b-b295-cb827dd154e7-0
00:05:53.960 --> 00:05:57.853
Well,
for cosine Theta to take a value of 0,

aee8d6a6-b94a-411b-b295-cb827dd154e7-1
00:05:57.853 --> 00:06:01.400
here's where my graph takes a value of 0.

3e99ce93-099b-4d70-a441-481e44420878-0
00:06:01.680 --> 00:06:06.802
And so I would get a solution here, here,
here, here, here, here,

3e99ce93-099b-4d70-a441-481e44420878-1
00:06:06.802 --> 00:06:10.760
and so on for cosine Theta to take a
value of 1/2.

33fc12e4-aea4-4743-8a1f-5870c3ce6af5-0
00:06:11.560 --> 00:06:15.600
I can think about how the function takes
a value of half at these points.

1bcfd38d-cf11-4dec-b8e1-d5229578f9f8-0
00:06:15.920 --> 00:06:21.865
And so as I go along here,
I would hit solutions when this dotted

1bcfd38d-cf11-4dec-b8e1-d5229578f9f8-1
00:06:21.865 --> 00:06:27.360
line hits the graph here, here, here,
here, here, and so on.

6f052672-f0ff-44ef-aed8-4d7fd5a19f14-0
00:06:27.760 --> 00:06:33.578
And similarly,
when cosine Theta equals one would be at

6f052672-f0ff-44ef-aed8-4d7fd5a19f14-1
00:06:33.578 --> 00:06:36.280
these maximum points here.

5f3cad7c-29a2-4bd6-939c-1c182978dd07-0
00:06:36.520 --> 00:06:44.200
So that would be when my dotted line for
one hits here, here and so on.

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So I can see a number of areas on the
graph where I am going to get solutions

9708cfd2-cb60-44c3-998a-1bbe3626b8c7-1
00:06:49.800 --> 00:06:55.400
and this is where I start to think about
wanting to have exactly 3 solutions.

4210d356-6764-44f0-ac68-0fc72a4df90c-0
00:06:55.640 --> 00:07:00.520
Now I'm I need to take the first 3 and so
that's going to be this one here,

4210d356-6764-44f0-ac68-0fc72a4df90c-1
00:07:00.520 --> 00:07:02.640
this one here, and this one here.

49ba9d5c-3143-4c84-8655-0cb70c38d2e0-0
00:07:03.720 --> 00:07:08.400
I don't want 4 solutions,
I want exactly 3 solutions.

a47d426b-0f3f-43d0-ac87-bc810aa61af5-0
00:07:08.640 --> 00:07:16.000
That means I won't want to take this one
or any any further solutions.

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00:07:16.000 --> 00:07:22.480
I get this solution when Theta is 0 and
so I want this interval to include zero,

14b6ec2e-1613-455c-ac6e-f7d4c65bc258-1
00:07:22.480 --> 00:07:26.560
but it already does which is helpful and
accurate.

36e305c8-75e5-44be-ac3c-6e7a599540bd-0
00:07:27.520 --> 00:07:34.231
I want to include all the way up to this
third solution and this solution happens

36e305c8-75e5-44be-ac3c-6e7a599540bd-1
00:07:34.231 --> 00:07:38.160
for a value of Theta at 90° and so my
interval.

b2fe271d-7b7e-46a5-8ca6-d48bd8f65a00-0
00:07:39.080 --> 00:07:43.200
This upper bound for my interval has to
include 90.

52d08cc7-42e2-4ae8-89c1-b76af8a23b8e-0
00:07:43.520 --> 00:07:49.560
So my value my range of values of X has
to start at and include 90.

f5bbe716-91e9-4187-9117-c13e1cdb5389-0
00:07:50.000 --> 00:07:58.280
Now X can then go all the way up to 270
but not include 270.

6087cd66-c4b7-4187-be95-8ed8d38d2b8f-0
00:07:58.640 --> 00:08:01.680
At 270 it would hit this 4th solution.

fc0f6800-7c07-4f55-804e-a1a4686931b9-0
00:08:01.880 --> 00:08:07.607
So I don't want the value of 270,
but I can go all the way up to 270 and

fc0f6800-7c07-4f55-804e-a1a4686931b9-1
00:08:07.607 --> 00:08:10.040
still have exactly 3 solutions.

82ac5375-7ae8-4e88-8092-d82347e27f93-0
00:08:10.240 --> 00:08:17.620
So the range of the the range of all
possible values of X is X between 90 and

82ac5375-7ae8-4e88-8092-d82347e27f93-1
00:08:17.620 --> 00:08:22.920
270 where 270 is not included and where
90 is included.

467fce7a-8c20-47e9-8f83-4164cbfce3f6-0
00:08:23.200 --> 00:08:27.597
Let's have a look at the options for us
to choose from to get our final answer

467fce7a-8c20-47e9-8f83-4164cbfce3f6-1
00:08:27.597 --> 00:08:28.600
for this question.

8a603fa3-a266-42e9-a802-9f800954b3e7-0
00:08:28.880 --> 00:08:32.560
And I can see that option B gives the
interval that we want.

12785f02-e8c5-47a5-98e4-a1f4e269ebda-0
00:08:32.560 --> 00:08:34.960
So B is the answer to this question.

2e70c445-db2a-4996-b1cd-ce956b11f92d-0
00:08:36.080 --> 00:08:38.560
Let's take a moment to reflect on this
question.

3fb79b88-a7f9-4274-aff3-3305e938298e-0
00:08:39.040 --> 00:08:40.880
This question was quite complex.

d615b614-351d-4066-a3a1-48d886028a50-0
00:08:40.880 --> 00:08:46.242
It had many stages to it,
many steps and many things to consider as

d615b614-351d-4066-a3a1-48d886028a50-1
00:08:46.242 --> 00:08:49.160
part of getting to that final answer.

80565c2c-d963-49e5-b249-bcd1a4760bee-0
00:08:49.440 --> 00:08:53.826
And this question to me felt like a
journey that when I started out,

80565c2c-d963-49e5-b249-bcd1a4760bee-1
00:08:53.826 --> 00:08:57.068
I didn't know how to describe the journey
in full,

80565c2c-d963-49e5-b249-bcd1a4760bee-2
00:08:57.068 --> 00:09:00.120
but at each junction I knew which road to
take.

f6b628d7-6eb9-4b83-bfb1-ed612037f172-0
00:09:00.360 --> 00:09:04.040
And sometimes questions feel like that,
at least they do for me anyway.

1e484e13-8ed8-4a9a-ab06-9815ceb4e7db-0
00:09:04.240 --> 00:09:08.622
So I started by taking something that was
very familiar to me,

1e484e13-8ed8-4a9a-ab06-9815ceb4e7db-1
00:09:08.622 --> 00:09:14.047
sine squared of a thing equals 3 / 4 and
I knew exactly what to do with that,

1e484e13-8ed8-4a9a-ab06-9815ceb4e7db-2
00:09:14.047 --> 00:09:19.960
but I didn't know what would happen later
on down the line until the maths came out.

b52aff5a-ba21-49b7-9850-65723842c5e6-0
00:09:20.640 --> 00:09:24.986
The other thing to consider for this
question is how many values do I take as

b52aff5a-ba21-49b7-9850-65723842c5e6-1
00:09:24.986 --> 00:09:25.600
I go along?

b12feca0-875f-4377-b03f-52b0463cf08f-0
00:09:25.840 --> 00:09:30.198
And the way I approach that is I
generally try and get a sense of taking

b12feca0-875f-4377-b03f-52b0463cf08f-1
00:09:30.198 --> 00:09:33.840
more values than I think I will need to
be on the safe side.

511369b4-1758-42d2-9b22-eb07d895a71c-0
00:09:35.400 --> 00:09:40.219
Something else important to consider for
this question is having a fluidity,

511369b4-1758-42d2-9b22-eb07d895a71c-1
00:09:40.219 --> 00:09:43.849
withdrawing your cosine graphs,
your sine graphs as well,

511369b4-1758-42d2-9b22-eb07d895a71c-2
00:09:43.849 --> 00:09:46.039
and knowing those reference angles.

4fe08c4d-d164-429c-865d-bba9bd07199b-0
00:09:46.040 --> 00:09:51.035
So the reference angle here for sine of
something being root 3 / 2,

4fe08c4d-d164-429c-865d-bba9bd07199b-1
00:09:51.035 --> 00:09:53.680
I needed to know that straight away.

c64fcf83-33b7-4d52-bfd2-e340013c919f-0
00:09:53.680 --> 00:09:58.230
And if you don't know those values,
then there are some triangles that you

c64fcf83-33b7-4d52-bfd2-e340013c919f-1
00:09:58.230 --> 00:09:59.080
can reference.

9c6d4c9d-0171-4be4-b9cb-c8899a619481-0
00:09:59.080 --> 00:09:59.960
I'll show you one of them.

43a8ad2c-1268-4b78-b47c-a359f277ff6a-0
00:10:00.240 --> 00:10:07.440
So the value of 60° came from an equal
triangle of side length 2.

e3e91d79-81a4-4756-b252-a7149b69a8e6-0
00:10:07.800 --> 00:10:11.567
And if I were to split that triangle
exactly in half,

e3e91d79-81a4-4756-b252-a7149b69a8e6-1
00:10:11.567 --> 00:10:16.520
I would get a right angled triangle with
a hypotenuse of 2A base of 1.

d6d52e9b-dc0d-42f4-a690-d4d2cd4085b6-0
00:10:16.680 --> 00:10:21.720
And this side length here by Pythagoras
would be square root 3.

ef4a03e8-53e1-4f5a-bd1d-73418eecb254-0
00:10:21.720 --> 00:10:25.120
Let me just draw in that right angle
there so you know which orientation it is.

fd342556-8000-40a2-a5dc-259836fd217c-0
00:10:26.280 --> 00:10:29.242
And because this came from an equilateral
triangle,

fd342556-8000-40a2-a5dc-259836fd217c-1
00:10:29.242 --> 00:10:33.686
this angle here is going to be 60° and
this angle here is going to be half of

fd342556-8000-40a2-a5dc-259836fd217c-2
00:10:33.686 --> 00:10:35.680
60° because I have about the angle.

b9865126-5726-4f52-be96-1047dcdfaa59-0
00:10:35.800 --> 00:10:37.240
So it's going to be an angle of 30.

b4c08119-eb62-4bf4-8588-29a541a3e15e-0
00:10:37.560 --> 00:10:45.361
And this triangle can be used to generate
values of cosine, sine,

b4c08119-eb62-4bf4-8588-29a541a3e15e-1
00:10:45.361 --> 00:10:48.080
and tan of 60° and 30°.

97752d0a-8187-4815-94cc-a7ece55bd68e-0
00:10:48.400 --> 00:10:53.162
So things like this are just helpful to
have in our mathematical toolkit as we

97752d0a-8187-4815-94cc-a7ece55bd68e-1
00:10:53.162 --> 00:10:56.840
approach exams like TAMIMA and this kind
of problem solving.