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Let's just do a ton of more examples, just so we
make sure that we're getting

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00:00:05,370 --> 00:00:07,020
this trig function thing down well.

3
00:00:07,020 --> 00:00:11,109
So let's construct ourselves some right triangles.

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Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

5
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it so far will only work in right triangles,
so if you try to find

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the trig

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00:00:20,579 --> 00:00:25,189
functions of angles that aren't part of right
triangles, we're goint to see a that we're going to have to construct right

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triangles, but let's just focus on the right triangles
for now.

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So let's say that I have a triangle, where
let's say this lengthy down here is seven,

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and let's say the length of this side up here, let's say that that is four.

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Let's figure out what the hypotenuse over here is going to be. So we know

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-let's call the hypotenuse "h"-

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00:00:45,350 --> 00:00:52,350
we know that h squared is going to be equal
to seven squared plus four squared, we know

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that from of the Pythagorean theorem.

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That the hypotenuse squared is equal to

16
00:00:57,190 --> 00:01:00,289
the square of each of the - the sum of the
squares

17
00:01:00,289 --> 00:01:04,370
of the other two sides. Eight squared is equal to seven
squared plus four squared.

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So this is the equal to forty-nine

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plus sixteen,

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fourty-nine plus sixteen,

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forty nine plus ten is fifty-nine, plus
six is

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sixty-five. It is a sixty five so this h squared,

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let me right: h squared,

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that's different shade of yellow, so we have h squared is equal to

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sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

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is sixty-five, or we could see that h is equal to, if we take the square root of

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both sides

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square root

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00:01:39,340 --> 00:01:42,850
square root of sixty five. And we really can simplify
this at all

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this is thirteen

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this is the same thing is thirteen times five,
both of those are not perfect squares and

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they're both

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prime so you can't simplify this any more.

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So this is equal to the square root

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of sixty five.

36
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Now let's find the trig functions for this angle
up here, let's call that angle up there theta.

37
00:02:05,060 --> 00:02:06,270
So whenever you do it

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00:02:06,270 --> 00:02:09,679
you always want to write down - at least for
me it works out to write down -

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00:02:09,679 --> 00:02:11,219
"soh cah toa".

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00:02:11,219 --> 00:02:12,829
soh...

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00:02:12,829 --> 00:02:16,429
...soh cah toa. I have these vague memories

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of my

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trigonometry teacher, maybe I've read it in some
book, I don't know - you know about

44
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some type of indian princess named "soh cah toa" or whatever, but it's a very useful

45
00:02:25,159 --> 00:02:28,079
new monic, so we can apply "soh cah toa" to find

46
00:02:28,079 --> 00:02:34,099
let's say we want to find the cosine, we want to find the cosine of our angle,

47
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we wanna find the cosine of our angle, you
say: "soh cah toa!"

48
00:02:37,779 --> 00:02:41,219
So the "cah" tells us what to do with cosine,

49
00:02:41,219 --> 00:02:43,070
the "cah" part tells us

50
00:02:43,070 --> 00:02:46,930
that cosine is adjacent over hypotenuse.

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00:02:46,930 --> 00:02:49,979
Cosine is equal to adjacent

52
00:02:49,979 --> 00:02:51,529
over hypotenuse.

53
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So let's look over here to theta; what side is adjacent?

54
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Well we know that the hypotenuse

55
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we know that that hypotenuse is this side over here

56
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so it can't be that side. The only
other side that's kind of adjacent to it that

57
00:03:04,579 --> 00:03:06,949
isn't the hypotenuse, is this four.

58
00:03:06,949 --> 00:03:10,479
So the adjacent side over here, that side is,

59
00:03:10,479 --> 00:03:14,149
it's right next to the angle, it's one of
the sides that kind of forms the angle

60
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it's four

61
00:03:15,269 --> 00:03:16,669
over the hypotenuse.

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The hypotenuse we already know, it's square root
of sixty-five, so it's four

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over

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the square root of sixty-five.

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And sometimes people will want you to rationalize
the denominator which means they don't like

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to have an irrational number in the denominator,
like the square root of sixty five

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and if they - if you wanna rewrite this without
the

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irrational number in the denominator, you can
multiply the numerator and the denominator

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by the square root of sixty-five.

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This clearly will not change the number, because we're multiplying it by something over itself, so we're

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00:03:47,409 --> 00:03:51,279
multiplying the number by one. That won't change
the number, but at least it gets rid of the

72
00:03:51,279 --> 00:03:54,539
irrational number in the denominator. So the numerator
becomes

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four times the square root of sixty-five,

74
00:03:57,749 --> 00:04:03,280
and denominator square two sixty five times
square to sixty-five is just going to be sixty-five.

75
00:04:03,280 --> 00:04:07,129
We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.

76
00:04:07,129 --> 00:04:09,629
now let's do the other trick functions

77
00:04:09,629 --> 00:04:13,219
carelessly said the court rick functions will
learn in the future that there's a few tumble

78
00:04:13,219 --> 00:04:15,249
but they're all derived from these

79
00:04:15,249 --> 00:04:19,889
fussing about the sign of state is once again
to go to so cut off

80
00:04:19,889 --> 00:04:25,650
the sole tells what to do it signed side of
the opposite over high positives

81
00:04:25,650 --> 00:04:26,300
sign

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00:04:26,300 --> 00:04:27,629
is equal to

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00:04:27,629 --> 00:04:31,509
opposite over hypothesize opposite over high
partners

84
00:04:31,509 --> 00:04:34,040
so for this and all tried as officer

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00:04:34,040 --> 00:04:35,129
we just go

86
00:04:35,129 --> 00:04:38,930
opposite itworld opens into it's opposite
the seven

87
00:04:38,930 --> 00:04:41,909
so the opposite side is the seven this is

88
00:04:41,909 --> 00:04:44,349
right here that is the opposite side

89
00:04:44,349 --> 00:04:45,710
and then in the

90
00:04:45,710 --> 00:04:51,039
hypothalamus opposite overabundance of hypothesis
in the square root of sixty-five

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00:04:51,039 --> 00:04:52,760
square root sixty-five

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and once again if we want to rationalize this
week a multiply times square in sixty five

93
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over the square to sixty-five

94
00:05:00,469 --> 00:05:06,500
numerator will get seven square roots of sixty-five
and in the denominator we will get jest

95
00:05:06,500 --> 00:05:08,089
sixty-five again

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00:05:08,089 --> 00:05:10,219
knowledge do tangent

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let us do ten gente

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00:05:12,479 --> 00:05:15,689
so if i ask you the tangent

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but ended up there

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00:05:17,330 --> 00:05:19,979
once again go back to soak up

101
00:05:19,979 --> 00:05:23,120
toa the toll apart tells us what to do a tangent

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it tells us

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it tells us that tangent

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is equal to opposite over adjacent is equal
to opposite

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over

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opposite over ed to use it

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so for this and all

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what is opposite we've already figured it
out it's evident opens into the seventh opposite

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the seven

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so it's seven

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over website is adjacent

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well this for is adjacent

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this for is adjacent to the jason side is
four

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so it's seven

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over

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and were dot

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we figured out all of the trade ratios for
baidoa let's do it other one

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let's do another one on the colluded concrete
'cause right now we've been saying oh was

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tentative actress tens of their lives make
a little bit more concrete

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let's say

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let's say i mean drawn other right triangle

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list another right triangle here

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everything we're dealing with

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these are going to be right triangles

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let's say the high partners

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has the length

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00:06:21,919 --> 00:06:24,440
let's say that the side over here

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00:06:24,440 --> 00:06:26,469
has length too

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00:06:26,469 --> 00:06:31,830
and let's say that this length overhears will
be to times the square root of three we can

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verify that this works

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00:06:33,559 --> 00:06:38,279
if you have this aside squared so yet let
me write down two times the square root of

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three squared

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00:06:40,039 --> 00:06:42,930
plus to squared is equal to what

134
00:06:42,930 --> 00:06:43,889
this is

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to there's going to be four times three

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four times three plus four

137
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and this is going to be equal to twelve plus
for the for the sixteen and sixteen is indeed

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00:06:54,619 --> 00:06:57,729
four square so this does he call for squared

139
00:06:57,729 --> 00:07:02,419
it does equal foursquare it satisfies the
protagonist hero

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00:07:02,419 --> 00:07:06,529
if you remember some of your work from thirty
sixty ninety triangles that you might have

141
00:07:06,529 --> 00:07:09,050
learned in geometry you might recognize that
this

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00:07:09,050 --> 00:07:13,030
is eight thirty sixty ninety triangle this
right here is our right angle i should have

143
00:07:13,030 --> 00:07:16,219
drawn it from the get go to show that this
is the right triangle

144
00:07:16,219 --> 00:07:20,210
this angle right over here is our thirty degree
angle

145
00:07:20,210 --> 00:07:24,430
and then disable appear the single appears
is

146
00:07:24,430 --> 00:07:26,019
sixty degree angle

147
00:07:26,019 --> 00:07:28,139
and it's a pretty sixteen ninety because

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00:07:28,139 --> 00:07:31,990
the side opposite the thirty degrees is half
of my partners

149
00:07:31,990 --> 00:07:36,650
and in the side opposite the sixty degrees
is a square to three times the other side

150
00:07:36,650 --> 00:07:38,280
that's not the hype on this

151
00:07:38,280 --> 00:07:41,910
so that's that we're not gonna the student
was peer review of thirty sixty ninety triangles

152
00:07:41,910 --> 00:07:43,110
although i just get it

153
00:07:43,110 --> 00:07:46,830
let's actually fifi and the trigger issues
for the different angles

154
00:07:46,830 --> 00:07:48,080
so if i were to ask you

155
00:07:48,080 --> 00:07:51,059
or if anyone would ask you what is

156
00:07:51,059 --> 00:07:54,389
what is the signed of thirty degrees

157
00:07:54,389 --> 00:07:58,520
and remember thirty degrees is one of the
eagles in this triangle but it would apply

158
00:07:58,520 --> 00:08:01,520
whenever you have a thirty degree angle and
you're dealing with the right rangel will

159
00:08:01,520 --> 00:08:04,970
have broader definitions of the future but
if you see a sign of thirty degrees

160
00:08:04,970 --> 00:08:10,099
hey this ain't gold right over here is thirty
degrees so i can use this right triangle

161
00:08:10,099 --> 00:08:12,849
and we should remember so catawba

162
00:08:12,849 --> 00:08:14,439
rewrite it so

163
00:08:14,439 --> 00:08:15,949
ka

164
00:08:15,949 --> 00:08:17,270
toa

165
00:08:17,270 --> 00:08:22,159
sign tells us more also tells us what to do
it signed side is opposite over hype rotten

166
00:08:22,159 --> 00:08:23,050
is

167
00:08:23,050 --> 00:08:26,199
sign up thirty degrees is the opposite side

168
00:08:26,199 --> 00:08:29,279
that is the opposite side which is to

169
00:08:29,279 --> 00:08:32,149
over the hard part is the hype on this year's
for

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00:08:32,149 --> 00:08:35,800
it is to ford's which is the same thing as
one-half

171
00:08:35,800 --> 00:08:39,020
five thirty degrees you'll see is always going
to be equal

172
00:08:39,020 --> 00:08:40,760
to one-half

173
00:08:40,760 --> 00:08:42,190
now what is

174
00:08:42,190 --> 00:08:43,910
the duck cosign

175
00:08:43,910 --> 00:08:45,980
what is the co-signed of

176
00:08:45,980 --> 00:08:47,160
thirty degrees

177
00:08:47,160 --> 00:08:49,969
once again go back to seoul cut off

178
00:08:49,969 --> 00:08:56,070
the cod tells us what to do with co-sign co-sign
isn't jay sent over high popped in his

179
00:08:56,070 --> 00:08:59,940
so for looking at the thirty degree angle
it's the end jason this right over here is

180
00:08:59,940 --> 00:09:01,639
adjacent it's right next to it

181
00:09:01,639 --> 00:09:02,960
it's not my pot news

182
00:09:02,960 --> 00:09:06,790
it's the adjacent over the high positive so
it's too

183
00:09:06,790 --> 00:09:08,779
square roots of three

184
00:09:08,779 --> 00:09:10,320
and jay sent

185
00:09:10,320 --> 00:09:11,300
over

186
00:09:11,300 --> 00:09:13,820
over that partners over four

187
00:09:13,820 --> 00:09:19,290
or if we simplify that we divide the new renowned
abide to this the square root of three

188
00:09:19,290 --> 00:09:20,780
over too

189
00:09:20,780 --> 00:09:23,200
finally let's do that ended

190
00:09:23,200 --> 00:09:25,880
that tangent

191
00:09:25,880 --> 00:09:27,850
thirty degrees

192
00:09:27,850 --> 00:09:29,179
we go back to soak up

193
00:09:29,179 --> 00:09:30,080
draw

194
00:09:30,080 --> 00:09:34,900
so cut off poet tells us what to do it entered
it's opposite over adjacent

195
00:09:34,900 --> 00:09:38,860
you go to the thirty degree angle to the so
we care about ten to twelve thirty

196
00:09:38,860 --> 00:09:42,760
tended of thirty opposite is too

197
00:09:42,760 --> 00:09:47,150
opposite is to and the adjacent is to squirts
of three it's right next to it it's adjacent

198
00:09:47,150 --> 00:09:47,830
to it

199
00:09:47,830 --> 00:09:49,430
and jason means next to

200
00:09:49,430 --> 00:09:51,720
so to square roots of three

201
00:09:51,720 --> 00:09:53,110
so this is equal to

202
00:09:53,110 --> 00:09:56,820
the tunes cancel out one over the square root
of three

203
00:09:56,820 --> 00:10:00,340
oregon multiplied the numerator denominator
by the square three

204
00:10:00,340 --> 00:10:01,740
so we have

205
00:10:01,740 --> 00:10:03,290
square root of three

206
00:10:03,290 --> 00:10:05,200
over square root of three

207
00:10:05,200 --> 00:10:09,600
and so this is going to be equal to the numerator
square the story and then the denominator

208
00:10:09,600 --> 00:10:14,900
right overhear is just going to be three so
thats we've rationalize a square two three

209
00:10:14,900 --> 00:10:15,890
over three

210
00:10:15,890 --> 00:10:16,720
fair enough

211
00:10:16,720 --> 00:10:20,500
welch is the same triangle figure out the
tree gracias for the sixty degrees

212
00:10:20,500 --> 00:10:23,200
century we've already brought it

213
00:10:23,200 --> 00:10:24,890
so what is

214
00:10:24,890 --> 00:10:30,580
what is in the sign of the sixty degrees and
interval for getting the hang of it now

215
00:10:30,580 --> 00:10:35,480
side is opposite over js insult from the sock
it all out for the sixty degree angle website

216
00:10:35,480 --> 00:10:36,760
is opposite

217
00:10:36,760 --> 00:10:42,920
what opens out into the two squares of three
so the opposite side is two squares history

218
00:10:42,920 --> 00:10:47,880
from the sixty degree angle d<u>j</u> course ties
observer had taught me a story confuse you

219
00:10:47,880 --> 00:10:54,420
so it's opposite over high pot and his sister
squirts of three over four force a high partners

220
00:10:54,420 --> 00:10:58,750
so it is in equal to the simplifies to square
root of three

221
00:10:58,750 --> 00:11:00,000
over too

222
00:11:00,000 --> 00:11:03,139
what is the the co-sign a sixty degrees

223
00:11:03,139 --> 00:11:05,580
co-sign of sixty degrees

224
00:11:05,580 --> 00:11:10,330
so remember so cut off co-sign is adjacent
over my pot news

225
00:11:10,330 --> 00:11:15,070
it jason's is the two sides right next to
sixty degree angle so it's too

226
00:11:15,070 --> 00:11:17,920
over the hype on news which is four

227
00:11:17,920 --> 00:11:19,900
so this is equal to

228
00:11:19,900 --> 00:11:20,860
one-half

229
00:11:20,860 --> 00:11:22,120
and then finally

230
00:11:22,120 --> 00:11:24,460
what is a bit engine

231
00:11:24,460 --> 00:11:26,000
what is the tangent

232
00:11:26,000 --> 00:11:27,830
of sixty degrees

233
00:11:27,830 --> 00:11:32,790
well tended so cut toa attended is opposite
over adjacent

234
00:11:32,790 --> 00:11:34,220
opposite the sixty degrees

235
00:11:34,220 --> 00:11:36,130
is to squirts of three

236
00:11:36,130 --> 00:11:37,940
to squirts of three

237
00:11:37,940 --> 00:11:39,570
and adjacent to that

238
00:11:39,570 --> 00:11:43,020
in j sent to that

239
00:11:43,020 --> 00:11:45,470
adjacent to sixty degrees is too so it's

240
00:11:45,470 --> 00:11:48,750
opposite over it isn't too squirts of three
over too

241
00:11:48,750 --> 00:11:52,630
which is just equal to the square read of

242
00:11:52,630 --> 00:11:56,000
and i just wanted you look at these are related
to sign a third agrees is the same things

243
00:11:56,000 --> 00:11:58,150
of course i love sixty degrees

244
00:11:58,150 --> 00:12:01,750
because i'm a third agrees with the same thing
as a sign of sixty degrees and that these

245
00:12:01,750 --> 00:12:04,980
guys are the inverse of each other i think
if you think of a little bit about the strike

246
00:12:04,980 --> 00:12:05,440
both

247
00:12:05,440 --> 00:12:09,519
will start to make sense why will keep extending
this may be a lot more practice in the next

248
00:12:09,519 --> 00:12:10,110
few videos