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509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:16:18
This does not follow from the defn of limit.
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209372b213bd479ca6cc76c57845c087
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dulg
02/05/2022 11:17:15
what is the difference between this and the defn of a limit?
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:17:24
where N is used, instead of N_0
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197,712,244,354,711,550
dulg
3377
dulg
false
509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:17:42
recall the defn of limit
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:18:03
Limit of sequence $$(\forall\varepsilon\in\bR^+)(\exists N\in\bN)\{a_n:n\geq N\}\subseteq(L-\varepsilon, L+\varepsilon)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:18:05
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:18:07
Or
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209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:18:38
ah, N is a set of numbers in \bN, and N_0 is a one particular number?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:18:56
$$(\forall\varepsilon\in\bR^+)(\exists N\in\bN)(\forall n\in\bN) (n \ge N\implies \abs{L - a_n} < \varepsilon)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:18:58
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:19:03
No, you misunderstand
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:19:18
You have probably seen this definition, correct
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:19:24
yep
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0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:19:51
Personally, I think this one is easier to visualize
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:19:52
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:20:03
The (L - e, L + e) represents an interval
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:20:27
In either case, you should realise the N_0 in the original question does not correspond to N
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:20:44
how come?
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0
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209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:20:59
Let me write.
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1
510,789,298,321,096,700
TeXit
0796
TeXit
true
509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:21:22
$$(\forall\varepsilon\in\bR^+)(\exists N_1\in\bN)(\forall n\in\bN) (n \ge N_1\implies \abs{s - s_n} < \varepsilon)$$ $$(\forall\varepsilon\in\bR^+)(\exists N_2\in\bN)(\forall n\in\bN) (n \ge N_2\implies \abs{t - t_n} < \varepsilon)$$
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:21:39
You should realise that the N's for different sequences are not necessarily the same
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:21:51
And moreover, then N0 in the original question has nothing to do with them
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:00
In fact --- the epsilons don't have to be the same either
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:11
But in this particular case, you can choose them to be the same if u like
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:22:22
I see
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:22:25
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:26
oops corrected
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:41
But what you CAN do
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:45
is construct a new N
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:22:49
that is helpful to you
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:23:14
I just looked at the answer for this exercise
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0
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:23:18
🤦
1
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:23:34
https://cdn.discordapp.c…8692/unknown.png
unknown.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:23:52
The N's are differently located,not necessarily the same
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:24:10
👍
1
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:24:13
(N1 and N2 depend on epsilon)
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:24:24
So you the answer says you should take max(N0, N1, N2)
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:24:26
yes?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:24:34
Then you can compare the inequalities
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:24:58
you wouldnt mind if i shared u what my textbook says, right?
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:25:11
https://cdn.discordapp.c…_at_00.23.25.png
Screen_Shot_2022-02-06_at_00.23.25.png
1
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:25:18
https://cdn.discordapp.c…8566/unknown.png
unknown.png
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:25:19
its 9.9c, btw
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:25:42
sorry for all of this
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:25:46
🤔
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:25:53
Your textbook doesn't even use epsilon-N to show
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:26:00
and uses some result instead
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:26:10
I mean it's honestly not hard to use epsilon-N
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209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:27:23
Yeah, but atleast he proves the results in a more pedagogical way
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:27:30
$$(\forall n\geq N_0) s_n \leq t_n$$ $$(\forall\varepsilon\in\bR^+)(\exists N_1\in\bN)(\forall n\ge N_1)(\abs{s - s_n} < \varepsilon)$$ $$(\forall\varepsilon\in\bR^+)(\exists N_2\in\bN)(\forall n\ge N_2) (\abs{t - t_n} < \varepsilon)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:27:47
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:27:48
You have these 3 things
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:27:59
You can combine them into one statement by setting a new N = max(N0, N1, N2)
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:28:27
$$(\forall\varepsilon\in\bR^+)(\exists N\in\bN)(\forall n\ge\bN)$$ $$(\abs{s - s_n} < \varepsilon \land \abs{t - t_n} < \varepsilon \land s_n \leq t_n)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:28:42
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:28:47
You work with just the inside
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:28:52
And you can use the triangle inequality
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:29:11
what does the wedge mean?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:29:12
$$\abs{s - s_n} < \varepsilon$$ $$\abs{t - t_n} < \varepsilon$$ $$s_n \leq t_n$$
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:29:17
logical and
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:29:25
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:29:28
So we have these 3 statements combined
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:30:51
$$s-t \leq t_n - s_n - t+s$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:31:13
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:31:16
This idea
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:31:34
$$s-t \leq (t_n - t) - (s_n - s)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:31:47
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
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509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:33:39
$$(t_n - t) - (s_n - s) \leq \abs {(t_n - t) - (s_n - s)}$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:33:41
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:34:07
I believe it's this? which would lead to the result
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:35:02
https://i.imgur.com/2UoEN1J.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:35:21
🤔
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:37:15
thanks @Shuri2060
1
0
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:37:23
I truly appreciate helping me out
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0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:37:24
no ive lost how to do it, but you should be able to show via this ^
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:37:26
👌
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:37:30
.close
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help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:20:34
How do you solve this question?
https://cdn.discordapp.c…4096/unknown.png
unknown.png
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0
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help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:20:53
Trigonometric stuff aren't really my thing
1
0
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help-10
6ba94cd173b14a14a606456644adec95
331,558,346,333,224,960
jan Niku
02/05/2022 12:24:43
which side will the largest angle be opposite
0
1
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:26:29
I don't understand?
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:26:39
So you're saying the opposite will be the largest angle?
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:34:37
never mind this question
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:35:18
do you mind explaining what SOHCAHTOA this is?
https://cdn.discordapp.c…3654/unknown.png
unknown.png
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:35:55
is it sin or csc? @Helpers
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:36:07
if i'm wrong please correct me.
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
420,244,167,726,333,950
kirby your mom
02/05/2022 12:36:58
first, draw the figure
0
1
452,459,964,041,134,100
| n O z | PILYONG PILYANG
6912
| n O z | PILYONG PILYANG
false
510
help-10
6ba94cd173b14a14a606456644adec95
521,326,998,195,273,700
shreksdad
02/05/2022 12:37:43
^^^^
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:40:22
done
https://cdn.discordapp.c…6446/unknown.png
unknown.png
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:40:30
drew the figure gents
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
420,244,167,726,333,950
kirby your mom
02/05/2022 12:41:11
great
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
521,326,998,195,273,700
shreksdad
02/05/2022 12:41:23
Complementary angles
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
420,244,167,726,333,950
kirby your mom
02/05/2022 12:41:30
https://cdn.discordapp.c…2220/unknown.png
unknown.png
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
521,326,998,195,273,700
shreksdad
02/05/2022 12:41:39
Or that
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
420,244,167,726,333,950
kirby your mom
02/05/2022 12:41:42
alternate interior angles*
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
521,326,998,195,273,700
shreksdad
02/05/2022 12:41:47
Both work
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
521,326,998,195,273,700
shreksdad
02/05/2022 12:41:48
🙂
0
1
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:42:27
this is csc right?
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
452,459,964,041,134,100
| n O z | PILYONG PILYANG
02/05/2022 12:42:31
if i'm correct
1
0
null
null
null
null
null
510
help-10
6ba94cd173b14a14a606456644adec95
420,244,167,726,333,950
kirby your mom
02/05/2022 12:42:37
tan
0
1
null
null
null
null
null