1
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Let's do a slightly more
involved Snell's law example.

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So I have this person
over here, sitting

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at the edge of this pool.

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And they have a little
laser pointer in their hand

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and they shine
their laser pointer.

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So in their hand,
where they shine,

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00:00:14,660 --> 00:00:18,970
it's 1.7 meters above
the surface of the pool.

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00:00:18,970 --> 00:00:21,980
And they shine it so
it travels 8.1 meters

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00:00:21,980 --> 00:00:24,600
to touch the surface
of the water.

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And then the light
gets refracted inward.

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00:00:27,660 --> 00:00:28,950
It's going to a slower medium.

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If you think about
the car analogy,

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the outside tires get to stay
outside a little bit longer,

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so they move faster.

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So it gets refracted inward.

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And then it hits the
bottom of the pool

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at some point right over here.

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And the pool, they tell
us, is three meters deep.

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What I want to figure out is how
far away does this point hit.

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So what is this distance
right over here?

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And to figure that
out, I just need

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to figure out what
this distance is.

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I need to figure out what this
distance is-- so this distance

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right over here.

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And then figure out what
that distance is, and then

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add them up.

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So I can figure out
this part-- trying

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to do it in a different
color-- this part right

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until we hit the
surface of the water

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and then figure out this
incremental distance, just

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like that.

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And hopefully with a
little trigonometry

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and maybe a little
bit of Snell's law

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00:01:21,310 --> 00:01:22,620
we'll be able to get there.

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So let's start on maybe
what's the simplest thing.

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Let's just figure
out this distance.

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00:01:26,610 --> 00:01:29,360
And it looks like it will
pay off later on, as well.

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So let's figure out this
distance right over here.

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00:01:31,950 --> 00:01:35,800
So just the distance along
the surface of the water,

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to where the laser
point actually

43
00:01:37,860 --> 00:01:39,810
starts touching the water.

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00:01:39,810 --> 00:01:42,760
And this is just a straight up
Pythagorean theorem problem.

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This is a right angle.

46
00:01:44,870 --> 00:01:46,540
This is the
hypotenuse, over here.

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So this distance, let's
call this distance x.

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x squared plus
1.7 meters squared

49
00:01:54,030 --> 00:01:56,227
is going to be equal
to 8.1 squared-- just

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00:01:56,227 --> 00:01:57,560
straight up Pythagorean theorem.

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00:01:57,560 --> 00:02:01,620
So x squared plus
1.7 squared is going

52
00:02:01,620 --> 00:02:04,260
to be equal to 8.1 squared.

53
00:02:04,260 --> 00:02:07,140
Or we could subtract 1.7
squared from both sides.

54
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We get x squared is equal to
8.1 squared minus 1.7 squared.

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If we want to solve
for x, x is going

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to be the positive
square root of this,

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because we only care
about positive distances.

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x is going to be equal to the
principal root of 8.1 squared

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00:02:24,570 --> 00:02:27,460
minus 1.7 squared.

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00:02:27,460 --> 00:02:29,210
And let's get our
calculator out for that.

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So x is going to be equal to
the square root of 8.1 squared

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00:02:38,190 --> 00:02:42,760
minus 1.7 squared.

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And I get 7.9-- looks about--
let me just round it, 7.92.

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So x is about 7.92, although we
could save that number there,

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00:02:53,900 --> 00:02:55,510
so we can get a
more exact number.

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00:02:55,510 --> 00:03:01,300
So this is equal to 7.92.

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That is x.

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00:03:02,440 --> 00:03:05,160
Now, we just have to figure out
this incremental distance right

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00:03:05,160 --> 00:03:07,860
over here, add that
to this x, and then we

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know this entire distance.

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So let's see how we
can think about it.

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00:03:11,210 --> 00:03:15,700
So let's think about what
the incident angle is

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and then the angle
of refraction is.

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So I've dropped a
perpendicular to the interface,

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or to the surface.

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00:03:22,950 --> 00:03:27,970
So our incident angle is
this angle right over here.

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That is our incident angle.

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00:03:30,750 --> 00:03:32,830
And remember,
Snell's law-- we care

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about the sine of this angle.

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Actually, let me just write
down what we want to care about.

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00:03:36,720 --> 00:03:40,140
So we know this is
our incident angle.

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00:03:40,140 --> 00:03:43,540
This is our angle of refraction.

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00:03:43,540 --> 00:03:46,000
We know that the
index of refraction

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00:03:46,000 --> 00:03:47,510
for this medium
out here-- and this

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00:03:47,510 --> 00:03:51,690
is air, so it's going to be the
index of refraction for air--

86
00:03:51,690 --> 00:03:57,790
times the sine of theta 1--
this is just Snell's law,

87
00:03:57,790 --> 00:04:00,200
so times our incident
angle, right here--

88
00:04:00,200 --> 00:04:03,350
is going to be equal to
the index of refraction

89
00:04:03,350 --> 00:04:06,910
for the water-- and
we'll put the values

90
00:04:06,910 --> 00:04:11,970
in the next step-- times
the sine of theta 2-- times

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00:04:11,970 --> 00:04:16,890
the sine of our refraction
angle, sine of theta 2.

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00:04:16,890 --> 00:04:20,100
Now, we know-- we can
figure out these ends

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from this table right over here.

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00:04:21,579 --> 00:04:24,220
I actually got this problem
from ck12.org's flex book

95
00:04:24,220 --> 00:04:27,300
as well or at least the
image for the problem.

96
00:04:27,300 --> 00:04:29,810
And so if we want to
solve for theta 2--

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00:04:29,810 --> 00:04:32,174
or if we know theta 2, we
could then solve for this.

98
00:04:32,174 --> 00:04:34,340
And we'll do that with a
little bit of trigonometry.

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00:04:34,340 --> 00:04:37,350
Actually we won't even have to--
if we know the sine of theta 2,

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we'll be able to solve for this.

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00:04:40,370 --> 00:04:42,270
All right, we'll think
about it either way.

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00:04:42,270 --> 00:04:44,840
Actually, we'll just
solve for this angle,

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00:04:44,840 --> 00:04:47,200
and then if we know
this angle, then we'll

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00:04:47,200 --> 00:04:49,810
be able to use a little
trigonometry to figure out

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00:04:49,810 --> 00:04:51,352
this distance over here.

106
00:04:51,352 --> 00:04:53,560
So to solve for that angle,
we can look up these two.

107
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And so we just have to
figure out what this is.

108
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We need to figure out what
the sine of theta 1 is.

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00:04:57,590 --> 00:05:00,820
So let's put in
all of the values.

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Our index of refraction
of air is 1.00029.

111
00:05:04,160 --> 00:05:05,770
So let me put that in there.

112
00:05:05,770 --> 00:05:08,140
So that's this.

113
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So 1.00029 times
the sine of theta.

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00:05:13,144 --> 00:05:15,310
And you say, oh, how do we
figure out sine of theta?

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We don't even know
what that angle is.

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But remember, this is
basic trigonometry.

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Remember "soh cah toa."

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Sine is opposite
over hypotenuse.

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So if you have this
angle here-- let's

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00:05:26,840 --> 00:05:30,920
make it a part of
a right triangle.

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00:05:30,920 --> 00:05:33,550
So if you make that as part
of a right triangle, opposite

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00:05:33,550 --> 00:05:35,950
over hypotenuse-- it's
the ratio of this side,

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00:05:35,950 --> 00:05:39,827
it's the ratio of that
distance to the hypotenuse.

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This distance over here
we just figured out,

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it's the same as this
distance down here.

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It's x.

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So this is 7.92.

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So the sine of theta
1 is going to be

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00:05:49,330 --> 00:05:53,050
the opposite of the angle,
opposite over the hypotenuse.

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00:05:53,050 --> 00:05:55,920
That just comes from
the definition of sine.

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So it's going to be times-- so
this part right over here, sine

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00:05:59,380 --> 00:06:01,930
of theta 1-- we don't have
to know what theta 1 is.

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It's going to be 7.92 over 8.1.

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And that's going to be equal
to the index of refraction

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of water.

136
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So that's index of
water is 1.33-- so let

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00:06:21,307 --> 00:06:22,640
me do that in a different color.

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00:06:22,640 --> 00:06:26,060
So that's going to be-- no, I
wanted to do a different color.

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So that's going to be-- let
me do it in this dark blue.

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So that's going to be 1.33
times sine of theta 2.

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00:06:35,851 --> 00:06:37,850
And so, if we want to
solve for sine of theta 2,

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00:06:37,850 --> 00:06:41,900
you just divide both sides
of this equation by 1.33.

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So let's do that.

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00:06:42,650 --> 00:06:43,650
So I'll do it over here.

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So if you divide
both sides by 1.33,

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00:06:46,420 --> 00:06:58,280
we get 1.00029
times 7.92 over 8.1,

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00:06:58,280 --> 00:07:00,880
and we're also going
to divide by 1.33.

148
00:07:00,880 --> 00:07:06,120
So we're also dividing by 1.33.

149
00:07:06,120 --> 00:07:08,940
That is going to be equal
to this sine of theta 2.

150
00:07:08,940 --> 00:07:12,920


151
00:07:12,920 --> 00:07:15,050
So let's figure what that is.

152
00:07:15,050 --> 00:07:16,210
So let's do that.

153
00:07:16,210 --> 00:07:19,650
Get the calculator out.

154
00:07:19,650 --> 00:07:26,174
So we have 1.00029 times 7.92.

155
00:07:26,174 --> 00:07:28,340
Well, actually I could even
say times second answer,

156
00:07:28,340 --> 00:07:30,250
if we want this exact value.

157
00:07:30,250 --> 00:07:32,750
That was the last-- so I'm going
to do that-- second answer.

158
00:07:32,750 --> 00:07:35,570
So that's the actual
precise, not even rounding.

159
00:07:35,570 --> 00:07:40,030
And then we want to divide by
1.33, that's this right here.

160
00:07:40,030 --> 00:07:45,460
And then we want to divide
by 8.1, and we get that.

161
00:07:45,460 --> 00:07:48,890
And that's going to be equal
to the sine of theta 2.

162
00:07:48,890 --> 00:07:51,997
So that's going to be equal
to the sine of theta 2.

163
00:07:51,997 --> 00:07:53,080
So let me write this down.

164
00:07:53,080 --> 00:08:00,752
So we have 0.735 is equal
to the sine of theta 2.

165
00:08:00,752 --> 00:08:02,210
Now, we could take
the inverse sine

166
00:08:02,210 --> 00:08:04,540
of both sides of this
equation to solve for theta 2.

167
00:08:04,540 --> 00:08:07,470
So we get theta 2 is
equal to-- let's just take

168
00:08:07,470 --> 00:08:10,340
the inverse sine of this value.

169
00:08:10,340 --> 00:08:14,080
So I take the inverse sine of
the value that we just had,

170
00:08:14,080 --> 00:08:16,510
so answer is just
our last answer.

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00:08:16,510 --> 00:08:19,570
And we have theta 2
being 47.3-- let's

172
00:08:19,570 --> 00:08:22,550
say rounded-- 47.34 degrees.

173
00:08:22,550 --> 00:08:29,200
So this is 47.34 degrees.

174
00:08:29,200 --> 00:08:34,299
So we were able to figure out
what theta 2 is, 47.34 degrees.

175
00:08:34,299 --> 00:08:36,590
So now we just have to use
a little bit of trigonometry

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00:08:36,590 --> 00:08:41,058
to actually figure out
this distance over here.

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00:08:41,058 --> 00:08:44,490
Now what trig ratio involves--
so we know this angle.

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00:08:44,490 --> 00:08:49,540
We want to figure out its
opposite side, to that angle.

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00:08:49,540 --> 00:08:51,420
And we know the
adjacent side-- we

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00:08:51,420 --> 00:08:52,900
know that this right here is 3.

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00:08:52,900 --> 00:08:56,030
So what trig identity deals
with opposite and adjacent?

182
00:08:56,030 --> 00:08:57,900
Well, tangent-- toa.

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00:08:57,900 --> 00:09:00,740
Tangent is opposite
over adjacent.

184
00:09:00,740 --> 00:09:05,230
So we know that the tangent
of this angle right over here

185
00:09:05,230 --> 00:09:11,130
of 47.34 degrees is going to
be equal to this opposite side

186
00:09:11,130 --> 00:09:15,110
over here-- so I'll
call that y-- is going

187
00:09:15,110 --> 00:09:18,750
to equal y over
our adjacent side.

188
00:09:18,750 --> 00:09:20,400
And that's just 3 meters.

189
00:09:20,400 --> 00:09:24,340
Or if we want to solve
for y, we multiply

190
00:09:24,340 --> 00:09:26,330
both sides of this
equation by 3.

191
00:09:26,330 --> 00:09:34,840
You get 3 times the tangent of
47.34 degrees is equal to y.

192
00:09:34,840 --> 00:09:36,890
So let's just get
the calculator out.

193
00:09:36,890 --> 00:09:41,430
So 3 times the tangent
of that 47.34 degrees--

194
00:09:41,430 --> 00:09:45,170
I'll use the exact answer--
3 times the tangent of that

195
00:09:45,170 --> 00:09:48,470
is 3.255.

196
00:09:48,470 --> 00:09:51,610
So this distance right here, y.

197
00:09:51,610 --> 00:09:52,990
And we're at the home stretch.

198
00:09:52,990 --> 00:09:58,660
y is equal to 3.255 meters.

199
00:09:58,660 --> 00:10:02,280
Now, our question was, what
is this total distance?

200
00:10:02,280 --> 00:10:04,900
So it's going to be
this distance, x,

201
00:10:04,900 --> 00:10:07,130
plus y, plus the 3.25.

202
00:10:07,130 --> 00:10:09,830
So x was 7.92.

203
00:10:09,830 --> 00:10:11,540
And I'll round here.

204
00:10:11,540 --> 00:10:16,090
So it's literally just going
to be 7.92 plus our answer

205
00:10:16,090 --> 00:10:16,650
just now.

206
00:10:16,650 --> 00:10:19,270


207
00:10:19,270 --> 00:10:21,652
So we get about 11.18.

208
00:10:21,652 --> 00:10:24,110
Or maybe if we want to really
round, or get the same number

209
00:10:24,110 --> 00:10:26,350
significant digits,
maybe 11.2 meters.

210
00:10:26,350 --> 00:10:29,550
I'll just say 11.18 meters.

211
00:10:29,550 --> 00:10:34,820
So this right here, the distance
that we wanted to figure out

212
00:10:34,820 --> 00:10:40,300
is the point on the bottom of
the pool where the actual laser

213
00:10:40,300 --> 00:10:43,280
pointer actually hits
the surface of the pool

214
00:10:43,280 --> 00:10:49,950
will be 11.18--
approximately, I'm

215
00:10:49,950 --> 00:10:54,320
rounding a little bit-- meters
from this edge of the pool

216
00:10:54,320 --> 00:10:54,880
right there.

217
00:10:54,880 --> 00:10:55,990
Anyway, hopefully you
found that useful.

218
00:10:55,990 --> 00:10:58,531
It's a little bit more involved
than the Snell's law problem.

219
00:10:58,531 --> 00:11:00,790
But really the hard part was
just in the trigonometry,

220
00:11:00,790 --> 00:11:03,080
recognizing that you didn't
have to know this angle,

221
00:11:03,080 --> 00:11:04,590
because you have
all the information

222
00:11:04,590 --> 00:11:05,726
for the sine of that angle.

223
00:11:05,726 --> 00:11:07,600
You could actually figure
out that angle now.

224
00:11:07,600 --> 00:11:09,430
Now that you know its
sine, you could figure out

225
00:11:09,430 --> 00:11:10,340
the inverse sine of that.

226
00:11:10,340 --> 00:11:11,589
But that's not even necessary.

227
00:11:11,589 --> 00:11:15,010
We know the sine of the angle
using basic trigonometry.

228
00:11:15,010 --> 00:11:17,250
We can use that and
Snell's law to figure out

229
00:11:17,250 --> 00:11:18,534
this angle right here.

230
00:11:18,534 --> 00:11:19,950
And then once you
know this angle,

231
00:11:19,950 --> 00:11:21,420
use a little bit
more trigonometry

232
00:11:21,420 --> 00:11:24,255
to figure out this
incremental distance.

233
00:11:24,255 --> 00:00:00,000