JayDS
Solve the following equations over [0°, 360°].
1) 2 sin (x − 60)° = 1
2) √2 cos(x+90)°+1 = 0
Show working out thanks.
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Callisto
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For question (1)
2 sin (x − 60)° = 1
Divide both sides by 2, what do you get?
JayDS
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sin(x-60)°=1/2
Callisto
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Yes. Next, take arcsine for both sides, what do you get?
JayDS
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sorry, what's arcsine?
JayDS
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opposite of sine?
Callisto
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arcsine is the inverse of sine function.
JayDS
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(x-60)°= sin-1 (1/2)
JayDS
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sin^-1
Callisto
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Yes. What does \(sin^{-1}\frac{1}{2}\) give you?
JayDS
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30
Callisto
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Yes. That's the angle in quadrant I, what about in quadrant IV?
JayDS
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360-30=330 deg
Chlorophyll
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Or it's the good habit to memorize 1/2 = sin 30
JayDS
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I do remember it actually.
Callisto
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Oh~
So.. you get
x - 60 = 30 or x-60 = 330
Can you solve the two?
JayDS
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x=90 or x=390
Callisto
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Ah!!! My mistake!!
The other angle should be in quadrant II...
Since sine is positive in quadrant I and II.. I'm sorry!!
Callisto
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I'm sorry... Just start it again!
For question 1,
2 sin (x − 60)° = 1
Divide both sides by 2, which you've done just now.
sin (x − 60)° = 1/2
Take arcsine for both sides, you get
x-60 = 30 or x-60 = ______ (____is the angle in quadrant II, since sine of a angle in quadrant I and II gives you a positive value)
JayDS
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kk
Callisto
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Can you work out the ___ ?
JayDS
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Would it be 90?
Callisto
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Hmm....
We're still on x-60 = (angle in quadrant II), right?
JayDS
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yeh but I'm a bit confused with your method.
JayDS
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on that step.
JayDS
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how I was taught to do it was to find the angle in the first quadrant first, in this case it is 90 I believe?
JayDS
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and then using the angle that you found you use 180-that angle found= to get the answer
Callisto
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Can we do it in this way?
Solve siny = 1/2, what is/ are y?
JayDS
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y= sin^-1 (1/2)
Callisto
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which is equal to?
JayDS
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30
Callisto
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and?
JayDS
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150?
Callisto
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Right!
So, you get y=30 or y=150, agree?
JayDS
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yeh
Callisto
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Now, back you your question. 2 sin (x − 60)° = 1
sin (x-60) = 1/2
So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?
JayDS
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x-60=30 or x-60=150
x=90 or x= 210
Callisto
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Now, it's done~
JayDS
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yeh but I'm still a bit confused with a part of it.
Callisto
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Which part?
JayDS
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near the last part?
Callisto
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What is it exactly?
JayDS
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"So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?"
JayDS
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wait, is it possible if u type out all the working out in logical order?
Callisto
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Oh... perhaps I should write it in this way.
2sin(x-60) = 1
Let y = x-60, then the equation becomes 2siny = 1
So, siny = 1/2
And as what you did,
y= 30 or y=150
So, replace y by x-60
x-60 = 30 or x-60 = 150
Solve x.
It should be clearer now. Please ask if you still have questions. And sorry for my poor presentation just now :(
Chlorophyll
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Or you can visualize the circle:
1/2 = sin30 = sin ( 180 - 30 )
JayDS
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kk, I get the working out but yeh I may not remember how to do it your way as I was sort of taught a different way by my cousin and I absolutely hate these type of confusing type questions.
Callisto
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It's okay. Just choose the best way for yourself. :)
Can you do question 2?
JayDS
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yep, well I'll try do it your way and I think I can only do half of the steps
Callisto
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You can try to do it using your own way first :)
Callisto
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I'll check it for you
JayDS
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wait, so I've done this so far using your method, the rest I don't get.
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2
let y = x + 90
therefore cos y = -1/√2
y = ............ or y = ............
Callisto
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So far so good :)
What are the values of y such that cosy = -1/√2
JayDS
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not sure
Chlorophyll
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Hint: -1/√2 = -√2/ 2
Callisto
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Hmm.. Forget my method...
Can you solve it using your OWN way??
JayDS
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well it was my cousin's way and he only taught me just then. I still don't quite fully understand his method that's why I was planning to see if anyone could simplify it down for me on here.
Callisto
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Can you do me a favour by showing how your cousin has taught you? You can use another question or simply this question if you like.
JayDS
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tan (x+45°) = 1 x∈[0°,360°]
x+45 = 45° (x+45°) ∈ [45°,405°]
180° + 45° or 405°
x+45° = 45°, 225° or 405°
x = 0°, 180° or 360°
Chlorophyll
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Your cousin's way is exactly as same as our way!
JayDS
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lol, really?
Callisto
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I hope you don't mind me asking the following question.
1. Have you learnt CAST ?
2. Do you know something like cos(180 - x) = -cosx
sin(180-x) = sinx and so on?
3. Are you familiar with the trigo functions of special angles (that are 0 , 30 , 45, 60, 90, and so on) ?
It's practically the same
JayDS
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yes, I have and I do know the trig functions that u mentioned
Callisto
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Oh great.
Sudden quiz. What is cos135?
JayDS
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in exact value or degrees?
Chlorophyll
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Hint: 135 < 180
Callisto
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135 is in degrees.
cos135 gives you an exact value.
JayDS
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-cos45 = -Sqrt2/2
Callisto
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Yes.
Callisto
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Can we go back to your question 2 now?
JayDS
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yes, please. It's late over here and I want to go to sleep lol.
JayDS
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but I want to figure out how to do it properly first so I can do the other ones tomorrow.
Callisto
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Don't worry, I have to do my quiz in an hour too :|
Here's your work!
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2
let y = x + 90
therefore cos y = -1/√2
Can you solve y now?
JayDS
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wait. let me see... and good look for your quiz.
JayDS
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well since the value is negative it has to be in the second or third quadrant?
Callisto
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Yes. Another hint given by @Chlorophyll before : -1/√2 = - (1/√2) x (√2/√2) = - √2/2
JayDS
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from there I'm a bit stuck as to how to solve for y.
Chlorophyll
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Consider y as X!
Callisto
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cosy = -√2/2
What is the value of angle such that cos(angle) = -√2/2?
Another hint: refer to the sudden quiz
JayDS
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yes, but how did you get the -√2/2?
Callisto
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Ratinoalization?!
-1/√2 = - (1/√2) x (√2/√2) = - √2/2
JayDS
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can't u cancel the top and bottom Sqrt 2 and so u get -1/Sqrt2
Callisto
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Oh yes, of course! But you need to know that cos45 = 1/√2
Chlorophyll
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-√2/2 looks more familiar to you or not?
JayDS
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yes, it is cos 45 right?
JayDS
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-cos 45 in this case*
Callisto
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-cos45 = cos(___ -45) = cos(____+45) ?
JayDS
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-cos45 = cos(180-45) = cos (180+45)?
Callisto
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Yes. So...
cosy = -1/√2
y = ... or y= ...?
JayDS
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wait... let me write something else down first.
Callisto
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Take your time~
Chlorophyll
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@JayDS @Callisto giant leap improvement here =)
JayDS
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lol, no way, I haven't improved at all but Callisto has.
Callisto
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What??!!! You haven't improved at all :'( ???
JayDS
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√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2
let y = x + 90
therefore cos y = -1/√2
cos y = - √2/2
hmm where to from here?
JayDS
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I guess I have slightly lol, I'm just really stupid.
Chlorophyll
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@JayDS Our goal here is to help YOU improved :)
Callisto
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No one is stupid except me, you can't be stupid than me:|
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2
let y = x + 90
So, the equation becomes cos y = -1/√2
Your duty: solve y. How...?!
Callisto
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*delete ''you can't be stupid than me''
JayDS
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haha thanks, I guess I'm just a logical person, like I things shown out clearly well in a way that I can understand of course and I'm slow at learning, well depends on what type of maths as well.
JayDS
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I like*
Chlorophyll
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@Callisto wanna me to take over so you have time with your quiz?
JayDS
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sorry for taking up your time guys and thanks for your help so far.
Callisto
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@Chlorophyll Ah... I forget my quiz :| I'll come back as soon as possible!!
Chlorophyll
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@Callisto Good luck, Sis! Not to be worried I promise to a good job here :)
Callisto
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@Chlorophyll Sis, I trust you :) Thanks too!!
JayDS
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what's with the "Sis" lol
Chlorophyll
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√2 cos(x+90)°+1 = 0
-> cos(x+90)° = -1 /√2 = -√2/2
Since cos value is negative, now replace ... into where -√2/2 is, can you!
JayDS
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sorry, I'm confused about the replacing part.
Chlorophyll
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Hint: the angle in the second and third quadrants
JayDS
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yep, I know that but... I'm unsure how to use that
JayDS
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second quadrant = pi - angle so it will be pi - sqrt2 /2
third quadrant = pi + angle so it will be pi + Sqrt 2/2
Is that correct?
Chlorophyll
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cos(x+90)° = -√2/2
cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)
Chlorophyll
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So you have 2 cases here!
Chlorophyll
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x+90° = 180-45 and .....
JayDS
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yeh, that is the same as what I said earlier? pi - sqrt 2/2 and pi + sqrt 2/2 I believe.
JayDS
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but then you would need to convert is back to degrees.
Chlorophyll
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It's incorrect to connect the degree ( pi) with the radian ( arc length )
pi - sqrt 2/2
JayDS
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kk, can you quickly explain this line "cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)"
JayDS
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oh wait, I think I got it.
Chlorophyll
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Don't pay attention to the left part yet!
Chlorophyll
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-√2/2 = -cos45
Now we wish to get rid of negative sign:
-cos45 = cos ( 180 - 45)
-cos45 = cos ( 180 + 45)
Chlorophyll
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That's what Callisto has been working so hard to show you the concept!
JayDS
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√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2
let y = x + 90
So, the equation becomes cos y = -1/√2
cos y = -√2/2
cos y = -cos 45
cos (x+90) = -cos 45
cos45 = cos ( 180 - 45)
cos45 = cos ( 180 + 45)
Chlorophyll
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Actually now you've done with the right side, plug it into the equation:
cos (x+90) = -cos 45
In the second quadrant:
cos (x+90) = cos ( 180 - 45)
-> x+90 = ....?
JayDS
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x + 90 = 135???
Chlorophyll
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-> x = ...?
JayDS
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45
Chlorophyll
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Similarly for the third quadrant case, can you do it yourself?
JayDS
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well if x=45 wouldn't it just be 180+45=225?
Chlorophyll
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cos (x+90) = cos ( 180 + 45)
-> x+90 = ....?
JayDS
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x + 90 = 225
x = 135
Chlorophyll
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I suggest that you don't jump around, as we carefully show you step by step, just follow it one by one, and practice it yourself from now on!
JayDS
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LOL, okay big sis ^^ I will practice tomorrow but it's getting late so I'm going to sleep now.
Chlorophyll
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Okie, the solution already saved into your profile. Feel free to use it as your study note!
JayDS
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really, how?
JayDS
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is that a new function?
Chlorophyll
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Click into your profile, you'll see all your questions and answers!
Chlorophyll
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You can't see your own questions and answer in your profile?
JayDS
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Maybe you can only do because you have already reached 99 smart points or something.
JayDS
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SmartScore*
Chlorophyll
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No, you can check anybody profiles!
JayDS
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yeh kk, found it, it was on the left side, I didn't see it over there and I thought you couldn't click anything there.
JayDS
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Well I g2g now Sis, thanks so much for the help and help me say thanks to Callisto as well.