1
00:00:00,859 --> 00:00:04,478
Let's just do a ton of more examples, just so we
make sure that we're getting

2
00:00:04,478 --> 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.

4
00:00:11,109 --> 00:00:15,299
Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

5
00:00:15,299 --> 00:00:19,829
it so far, this will only work in right triangles,
so if you're trying to find

6
00:00:19,829 --> 00:00:24,204
the trig functions of angles that aren't part of right triangles, we're going to see that we're going to

7
00:00:24,204 --> 00:00:28,079
have to construct right triangles, but let's just focus on the right triangles for now.

8
00:00:28,079 --> 00:00:33,470
So let's say that I have a triangle, where
let's say this length down here is seven,

9
00:00:33,470 --> 00:00:39,190
and let's say the length of this side up here, let's say that that is four.

10
00:00:39,190 --> 00:00:43,170
Let's figure out what the hypotenuse over here is going to be. So we know

11
00:00:43,170 --> 00:00:45,350
-let's call the hypotenuse "h"-

12
00:00:45,350 --> 00:00:52,750
we know that h squared is going to be equal
to seven squared plus four squared, we know

13
00:00:52,750 --> 00:00:55,110
that from of the Pythagorean theorem,

14
00:00:55,110 --> 00:00:57,190
that the hypotenuse squared is equal to

15
00:00:57,190 --> 00:01:00,289
the square of each of the sum of the squares

16
00:01:00,289 --> 00:01:04,370
of the other two sides. Eight squared is equal to seven
squared plus four squared.

17
00:01:04,370 --> 00:01:08,147
So this is equal to forty-nine

18
00:01:08,147 --> 00:01:09,729
plus sixteen,

19
00:01:09,729 --> 00:01:11,740
forty-nine plus sixteen,

20
00:01:11,740 --> 00:01:16,851
forty nine plus ten is fifty-nine, plus
six is

21
00:01:16,851 --> 00:01:21,979
sixty-five. It is sixty five so this h squared,

22
00:01:21,979 --> 00:01:23,909
let me write: h squared

23
00:01:23,909 --> 00:01:28,310
-that's different shade of yellow- so we have h squared is equal to

24
00:01:28,310 --> 00:01:32,480
sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

25
00:01:32,480 --> 00:01:36,648
is sixty-five, or we could say that h is equal to, if we take the square root of

26
00:01:36,648 --> 00:01:38,120
both sides

27
00:01:38,120 --> 00:01:39,340
square root

28
00:01:39,340 --> 00:01:42,850
square root of sixty five. And we really can't simplify
this at all

29
00:01:42,850 --> 00:01:44,350
this is thirteen

30
00:01:44,350 --> 00:01:48,819
this is the same thing as thirteen times five,
both of those are not perfect squares and

31
00:01:48,819 --> 00:01:51,860
they're both prime so you can't simplify this any more.

32
00:01:51,860 --> 00:01:54,673
So this is equal to the square root

33
00:01:54,673 --> 00:01:56,248
of sixty five.

34
00:01:56,248 --> 00:02:04,956
Now let's find the trig, let's find the trig functions for this angle
up here. Let's call that angle up there theta.

35
00:02:05,060 --> 00:02:06,270
So whenever you do it

36
00:02:06,270 --> 00:02:09,679
you always want to write down - at least for
me it works out to write down -

37
00:02:09,679 --> 00:02:11,219
"soh cah toa".

38
00:02:11,219 --> 00:02:12,829
soh...

39
00:02:12,829 --> 00:02:16,429
...soh cah toa. I have these vague memories

40
00:02:16,429 --> 00:02:17,919
of my

41
00:02:17,919 --> 00:02:21,029
trigonometry teacher, maybe I've read it in some
book, I don't know - you know, some, about

42
00:02:21,029 --> 00:02:25,159
some type of indian princess named "soh cah toa" or whatever, but it's a very useful

43
00:02:25,159 --> 00:02:28,079
mnemonic, so we can apply "soh cah toa". Let's find

44
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.

45
00:02:34,099 --> 00:02:37,779
we wanna find the cosine of our angle, you
say: "soh cah toa!"

46
00:02:37,779 --> 00:02:41,219
So the "cah". "Cah" tells us what to do with cosine,

47
00:02:41,219 --> 00:02:43,070
the "cah" part tells us

48
00:02:43,070 --> 00:02:46,930
that cosine is adjacent over hypotenuse.

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

50
00:02:49,979 --> 00:02:51,529
over hypotenuse.

51
00:02:51,529 --> 00:02:55,909
So let's look over here to theta; what side is adjacent?

52
00:02:55,909 --> 00:02:57,759
Well we know that the hypotenuse

53
00:02:57,759 --> 00:03:00,639
we know that that hypotenuse is this side over here

54
00:03:00,639 --> 00:03:04,579
so it can't be that side. The only other side that's kind of adjacent to it that

55
00:03:04,579 --> 00:03:06,949
isn't the hypotenuse, is this four.

56
00:03:06,949 --> 00:03:10,479
So the adjacent side over here, that side is,

57
00:03:10,479 --> 00:03:14,149
it's literally right next to the angle, it's one of
the sides that kind of forms the angle

58
00:03:14,149 --> 00:03:15,269
it's four

59
00:03:15,269 --> 00:03:16,669
over the hypotenuse.

60
00:03:16,669 --> 00:03:21,929
The hypotenuse we already know is square root
of sixty-five, so it's four

61
00:03:21,929 --> 00:03:22,470
over

62
00:03:22,470 --> 00:03:25,130
the square root of sixty-five.

63
00:03:25,130 --> 00:03:29,460
And sometimes people will want you to rationalize
the denominator which means they don't like

64
00:03:29,460 --> 00:03:34,049
to have an irrational number in the denominator,
like the square root of sixty five

65
00:03:34,049 --> 00:03:37,559
and if they - if you wanna rewrite this without
a

66
00:03:37,559 --> 00:03:41,179
irrational number in the denominator, you can
multiply the numerator and the denominator

67
00:03:41,179 --> 00:03:43,049
by the square root of sixty-five.

68
00:03:43,049 --> 00:03:47,409
This clearly will not change the number, because we're multiplying it by something over itself, so we're

69
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

70
00:03:51,279 --> 00:03:54,539
irrational number in the denominator. So the numerator
becomes

71
00:03:54,539 --> 00:03:57,749
four times the square root of sixty-five,

72
00:03:57,749 --> 00:04:03,280
and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.

73
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.

74
00:04:07,129 --> 00:04:09,629
Now let's do the other trig functions

75
00:04:09,629 --> 00:04:13,219
or at least the other core trig functions. We'll
learn in the future that there's a ton of them

76
00:04:13,219 --> 00:04:15,249
but they're all derived from these

77
00:04:15,249 --> 00:04:19,889
so let's think about what the sign of theta is. Once again
go to "soh cah toa"

78
00:04:19,889 --> 00:04:25,650
the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

79
00:04:25,650 --> 00:04:27,383
Sine is equal to

80
00:04:27,383 --> 00:04:31,509
opposite over hypotenuse. Sine is opposite over hypotenuse.

81
00:04:31,509 --> 00:04:34,021
So for this angle what side is opposite?

82
00:04:34,021 --> 00:04:38,930
We just go opposite it, what it opens into, it's opposite
the seven

83
00:04:38,930 --> 00:04:41,909
so the opposite side is the seven.

84
00:04:41,909 --> 00:04:44,349
This right here - that is the opposite side

85
00:04:44,349 --> 00:04:45,710
and then in the

86
00:04:45,710 --> 00:04:50,008
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

87
00:04:50,008 --> 00:04:52,760
square root of sixty-five

88
00:04:52,760 --> 00:04:57,989
and once again if we wanted to rationalize this,
we could multiply times the square root of sixty-five

89
00:04:57,989 --> 00:05:00,469
over the square root of sixty-five

90
00:05:00,469 --> 00:05:06,500
and the the numerator, we'll get seven square root of sixty-five
and in the denominator we will get just

91
00:05:06,500 --> 00:05:08,089
sixty-five again.

92
00:05:08,089 --> 00:05:10,219
Now let's do tangent!

93
00:05:10,219 --> 00:05:12,479
Let us do tangent.

94
00:05:12,479 --> 00:05:15,551
So if i ask you the tangent

95
00:05:15,551 --> 00:05:17,330
of - the tangent of theta

96
00:05:17,330 --> 00:05:19,979
once again go back to soh cah

97
00:05:19,979 --> 00:05:23,120
toa the toa part tells us what to do a tangent

98
00:05:23,120 --> 00:05:24,550
it tells us

99
00:05:24,550 --> 00:05:27,319
it tells us that tangent

100
00:05:27,319 --> 00:05:31,989
is equal to opposite over adjacent is equal
to opposite

101
00:05:31,989 --> 00:05:33,379
over

102
00:05:33,379 --> 00:05:35,639
opposite over adjacent

103
00:05:35,639 --> 00:05:36,970
so for this angle

104
00:05:36,970 --> 00:05:41,380
what is opposite we've already figured it
out it's seven it opens into the seventh opposite

105
00:05:41,380 --> 00:05:42,549
the seven

106
00:05:42,549 --> 00:05:44,409
so it's seven

107
00:05:44,409 --> 00:05:46,089
over what side is adjacent

108
00:05:46,089 --> 00:05:48,009
well this four is adjacent

109
00:05:48,009 --> 00:05:51,040
this four is adjacent so the adjacent side is
four

110
00:05:51,040 --> 00:05:52,639
so it's seven

111
00:05:52,639 --> 00:05:54,049
over four

112
00:05:54,049 --> 00:05:54,950
and we're done

113
00:05:54,950 --> 00:05:59,349
we figured out all of the trig ratios for
theta let's do another one

114
00:05:59,349 --> 00:06:03,129
let's do another one. i'll make it a little bit concrete
'cause right now we've been saying oh was

115
00:06:03,129 --> 00:06:06,879
tangent of x, tangent of theta. let's make it a little bit more concrete

116
00:06:06,879 --> 00:06:08,310
let's say

117
00:06:08,310 --> 00:06:11,059
let's say, let me draw another right triangle

118
00:06:11,059 --> 00:06:13,999
that's another right triangle here

119
00:06:13,999 --> 00:06:15,250
everything we're dealing with

120
00:06:15,250 --> 00:06:18,110
these are going to be right triangles

121
00:06:18,110 --> 00:06:19,650
let's say the hypotenuse

122
00:06:19,650 --> 00:06:21,919
has length four

123
00:06:21,919 --> 00:06:24,440
let's say that this side over here

124
00:06:24,440 --> 00:06:26,469
has length two

125
00:06:26,469 --> 00:06:31,830
and let's say that this length over here is goint to be two times the square root of three we can

126
00:06:31,830 --> 00:06:33,559
verify that this works

127
00:06:33,559 --> 00:06:38,279
if you have this side squared so you have let
me write it down two times the square root of

128
00:06:38,279 --> 00:06:40,039
three squared

129
00:06:40,039 --> 00:06:42,930
plus two squared is equal to what

130
00:06:42,930 --> 00:06:43,889
this is

131
00:06:43,889 --> 00:06:47,119
two there's going to be four times three

132
00:06:47,119 --> 00:06:49,549
four times three plus four

133
00:06:49,549 --> 00:06:54,619
and this is going to be equal to twelve plus
four is equal to sixteen and sixteen is indeed

134
00:06:54,619 --> 00:06:57,729
four squared so this does equal four squared

135
00:06:57,729 --> 00:07:02,419
it does equal four squared it satisfies the pythagorean theorem

136
00:07:02,419 --> 00:07:06,529
and if you remember some of your work from thirty
sixty ninety triangles that you might have

137
00:07:06,529 --> 00:07:09,050
learned in geometry you might recognize that
this

138
00:07:09,050 --> 00:07:13,030
is a thirty sixty ninety triangle this
right here is our right angle i should have

139
00:07:13,030 --> 00:07:16,219
drawn it from the get go to show that this
is a right triangle

140
00:07:16,219 --> 00:07:20,210
this angle right over here is our thirty degree
angle

141
00:07:20,210 --> 00:07:24,430
and then this angle up here, this angle up here
is

142
00:07:24,430 --> 00:07:26,019
a sixty degree angle

143
00:07:26,019 --> 00:07:28,139
and it's a thirty sixteen ninety because

144
00:07:28,139 --> 00:07:31,990
the side opposite the thirty degrees is half the hypotenuse

145
00:07:31,990 --> 00:07:36,650
and then the side opposite the sixty degrees
is a squared three times the other side

146
00:07:36,650 --> 00:07:38,280
that's not the hypotenuse

147
00:07:38,280 --> 00:07:41,910
so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

148
00:07:41,910 --> 00:07:43,110
although i just did it

149
00:07:43,110 --> 00:07:46,830
let's actually find the trig ratios
for the different angles

150
00:07:46,830 --> 00:07:48,080
so if i were to ask you

151
00:07:48,080 --> 00:07:51,059
or if anyone were to ask you what is

152
00:07:51,059 --> 00:07:54,389
what is the sine of thirty degrees

153
00:07:54,389 --> 00:07:58,520
and remember thirty degrees is one of the
angles in this triangle but it would apply

154
00:07:58,520 --> 00:08:01,520
whenever you have a thirty degree angle and
you're dealing with the right triangle we'll

155
00:08:01,520 --> 00:08:04,970
have broader definitions in the future but
if you say sine of thirty degrees

156
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

157
00:08:10,099 --> 00:08:12,849
and we just have to remember soh cah toa

158
00:08:12,849 --> 00:08:14,439
rewrite it so

159
00:08:14,439 --> 00:08:15,949
cah

160
00:08:15,949 --> 00:08:17,270
toa

161
00:08:17,270 --> 00:08:22,159
sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

162
00:08:23,050 --> 00:08:26,199
sine of thirty degrees is the opposite side

163
00:08:26,199 --> 00:08:29,279
that is the opposite side which is two

164
00:08:29,279 --> 00:08:32,149
over the hypotenuse. the hypotenuse here is four.

165
00:08:32,149 --> 00:08:35,800
it is two fourths which is the same thing as
one-half

166
00:08:35,800 --> 00:08:39,020
sine of thirty degrees you'll see is always going
to be equal

167
00:08:39,020 --> 00:08:40,760
to one-half

168
00:08:40,760 --> 00:08:42,190
now what is

169
00:08:42,190 --> 00:08:43,910
the cosine

170
00:08:43,910 --> 00:08:45,980
what is the cosine of

171
00:08:45,980 --> 00:08:47,160
thirty degrees

172
00:08:47,160 --> 00:08:49,969
once again go back to soh cah toa.

173
00:08:49,969 --> 00:08:56,070
the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

174
00:08:56,070 --> 00:08:59,940
so for looking at the thirty degree angle
it's the adjacent this right over here is

175
00:08:59,940 --> 00:09:01,639
adjacent it's right next to it

176
00:09:01,639 --> 00:09:02,960
it's not the hypotenuse

177
00:09:02,960 --> 00:09:06,790
it's the adjacent over the hypotenuse so
it's two

178
00:09:06,790 --> 00:09:08,779
square roots of three

179
00:09:08,779 --> 00:09:10,320
adjacent

180
00:09:10,320 --> 00:09:11,300
over

181
00:09:11,300 --> 00:09:13,820
over the hypotenuse over four

182
00:09:13,820 --> 00:09:19,290
or if we simplify that we divide the numerator and the denominator by two it's the square root of three

183
00:09:19,290 --> 00:09:20,780
over two

184
00:09:20,780 --> 00:09:23,200
finally let's do

185
00:09:23,200 --> 00:09:25,880
the tangent

186
00:09:25,880 --> 00:09:27,850
tangent of thirty degrees

187
00:09:27,850 --> 00:09:29,179
we go back to soh cah toa

188
00:09:29,179 --> 00:09:30,080
soh cah toa

189
00:09:30,080 --> 00:09:34,900
toa tells us what to do with tangent
it's opposite over adjacent

190
00:09:34,900 --> 00:09:38,860
you go to the thirty degree angle because that's what we care about, tangent of thirty

191
00:09:38,860 --> 00:09:42,760
tangent of thirty opposite is two

192
00:09:42,760 --> 00:09:47,150
opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

193
00:09:47,150 --> 00:09:47,830
to it

194
00:09:47,830 --> 00:09:49,430
adjacent means next to

195
00:09:49,430 --> 00:09:51,720
so two square roots of three

196
00:09:51,720 --> 00:09:53,110
so this is equal to

197
00:09:53,110 --> 00:09:56,820
the twos cancel out one over the square root
of three

198
00:09:56,820 --> 00:10:00,340
or we could multiply the numerator and the denominator
by the square root of three

199
00:10:00,340 --> 00:10:01,740
so we have

200
00:10:01,740 --> 00:10:03,290
square root of three

201
00:10:03,290 --> 00:10:05,200
over square root of three

202
00:10:05,200 --> 00:10:09,600
and so this is going to be equal to the numerator
square root of three and then the denominator

203
00:10:09,600 --> 00:10:14,900
right over here is just going to be three so
thats we've rationalized a square root of three

204
00:10:14,900 --> 00:10:15,890
over three

205
00:10:15,890 --> 00:10:16,720
fair enough

206
00:10:16,720 --> 00:10:20,500
now lets use the same triangle to figure out the
trig ratios for the sixty degrees

207
00:10:20,500 --> 00:10:23,200
since we've already drawn it

208
00:10:23,200 --> 00:10:24,890
so what is

209
00:10:24,890 --> 00:10:30,580
what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

210
00:10:30,580 --> 00:10:35,480
sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side


211
00:10:35,480 --> 00:10:36,760
is opposite

212
00:10:36,760 --> 00:10:42,920
what opens out into the two square roots of three
so the opposite side is two square roots of three

213
00:10:42,920 --> 00:10:47,880
and from the sixty degree angle the adj-oh sorry its the

214
00:10:47,880 --> 00:10:54,420
opposite over hypotenuse, don't want to confuse you.

215
00:10:54,420 --> 00:10:58,750
so it is opposite over hypotenuse

216
00:10:58,750 --> 00:11:00,000
so it's two square roots of three over four. four is the hypotenuse.

217
00:11:00,000 --> 00:11:03,139
so it is equal to, this simplifies to square root of three over two.

218
00:11:03,139 --> 00:11:05,580
what is the cosine of sixty degrees. cosine of sixty degrees.

219
00:11:05,580 --> 00:11:10,330
so remember soh cah toa. cosine is adjacent over hypotenuse.

220
00:11:10,330 --> 00:11:15,070
adjacent is the two sides right next to the sixty degree angle so it's two

221
00:11:15,070 --> 00:11:17,920
over the hypotenuse which is four

222
00:11:17,920 --> 00:11:19,900
so this is equal to

223
00:11:19,900 --> 00:11:20,860
one-half

224
00:11:20,860 --> 00:11:22,120
and then finally

225
00:11:22,120 --> 00:11:24,460
what is the tangent, what is the tangent

226
00:11:26,000 --> 00:11:27,830
of sixty degrees

227
00:11:27,830 --> 00:11:32,790
well tangent soh cah toa tangent is opposite
over adjacent

228
00:11:32,790 --> 00:11:34,220
opposite the sixty degrees

229
00:11:34,220 --> 00:11:36,130
is two square roots of three

230
00:11:36,130 --> 00:11:37,940
two square roots of three

231
00:11:37,940 --> 00:11:39,570
and adjacent to that

232
00:11:39,570 --> 00:11:43,020
adjacent to that

233
00:11:43,020 --> 00:11:45,470
is two

234
00:11:45,470 --> 00:11:48,750
adjacent to sixty degrees is two

235
00:11:48,750 --> 00:11:52,630
so its opposite over adjacent

236
00:11:52,630 --> 00:11:56,000
two square roots of three over two which is just equal to

237
00:11:56,000 --> 00:11:58,150
the square root of three

238
00:11:58,150 --> 00:12:01,750
And I just wanted to - look how these are related

239
00:12:01,750 --> 00:12:03,365
the sine of thirty degrees is the same as the cosine of sixty degrees

240
00:12:03,365 --> 00:12:04,980
and then these guys are the inverse of each other and i think if you think a little bit about this triangle

241
00:12:05,440 --> 00:12:09,519
it will start to make sense why. we'll keep extending
this and give you a lot more practice in the next

242
00:12:09,519 --> 00:12:10,110
few videos