anonymous
  • anonymous
Solve the following equations over [0°, 360°]. 1) 2 sin (x − 60)° = 1 2) √2 cos(x+90)°+1 = 0 Show working out thanks.
Mathematics
chestercat
  • chestercat
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Callisto
  • Callisto
For question (1) 2 sin (x − 60)° = 1 Divide both sides by 2, what do you get?
anonymous
  • anonymous
sin(x-60)°=1/2
Callisto
  • Callisto
Yes. Next, take arcsine for both sides, what do you get?

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anonymous
  • anonymous
sorry, what's arcsine?
anonymous
  • anonymous
opposite of sine?
Callisto
  • Callisto
arcsine is the inverse of sine function.
anonymous
  • anonymous
(x-60)°= sin-1 (1/2)
anonymous
  • anonymous
sin^-1
Callisto
  • Callisto
Yes. What does \(sin^{-1}\frac{1}{2}\) give you?
anonymous
  • anonymous
30
Callisto
  • Callisto
Yes. That's the angle in quadrant I, what about in quadrant IV?
anonymous
  • anonymous
360-30=330 deg
anonymous
  • anonymous
Or it's the good habit to memorize 1/2 = sin 30
anonymous
  • anonymous
I do remember it actually.
Callisto
  • Callisto
Oh~ So.. you get x - 60 = 30 or x-60 = 330 Can you solve the two?
anonymous
  • anonymous
x=90 or x=390
Callisto
  • Callisto
Ah!!! My mistake!! The other angle should be in quadrant II... Since sine is positive in quadrant I and II.. I'm sorry!!
Callisto
  • Callisto
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)
anonymous
  • anonymous
kk
Callisto
  • Callisto
Can you work out the ___ ?
anonymous
  • anonymous
Would it be 90?
Callisto
  • Callisto
Hmm.... We're still on x-60 = (angle in quadrant II), right?
anonymous
  • anonymous
yeh but I'm a bit confused with your method.
anonymous
  • anonymous
on that step.
anonymous
  • anonymous
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?
anonymous
  • anonymous
and then using the angle that you found you use 180-that angle found= to get the answer
Callisto
  • Callisto
Can we do it in this way? Solve siny = 1/2, what is/ are y?
anonymous
  • anonymous
y= sin^-1 (1/2)
Callisto
  • Callisto
which is equal to?
anonymous
  • anonymous
30
Callisto
  • Callisto
and?
anonymous
  • anonymous
150?
Callisto
  • Callisto
Right! So, you get y=30 or y=150, agree?
anonymous
  • anonymous
yeh
Callisto
  • Callisto
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?
anonymous
  • anonymous
x-60=30 or x-60=150 x=90 or x= 210
Callisto
  • Callisto
Now, it's done~
anonymous
  • anonymous
yeh but I'm still a bit confused with a part of it.
Callisto
  • Callisto
Which part?
anonymous
  • anonymous
near the last part?
Callisto
  • Callisto
What is it exactly?
anonymous
  • anonymous
"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?"
anonymous
  • anonymous
wait, is it possible if u type out all the working out in logical order?
Callisto
  • Callisto
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 :(
anonymous
  • anonymous
Or you can visualize the circle: 1/2 = sin30 = sin ( 180 - 30 )
anonymous
  • anonymous
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
  • Callisto
It's okay. Just choose the best way for yourself. :) Can you do question 2?
anonymous
  • anonymous
yep, well I'll try do it your way and I think I can only do half of the steps
Callisto
  • Callisto
You can try to do it using your own way first :)
Callisto
  • Callisto
I'll check it for you
anonymous
  • anonymous
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
  • Callisto
So far so good :) What are the values of y such that cosy = -1/√2
anonymous
  • anonymous
not sure
anonymous
  • anonymous
Hint: -1/√2 = -√2/ 2
Callisto
  • Callisto
Hmm.. Forget my method... Can you solve it using your OWN way??
anonymous
  • anonymous
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
  • Callisto
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.
anonymous
  • anonymous
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°
anonymous
  • anonymous
Your cousin's way is exactly as same as our way!
anonymous
  • anonymous
lol, really?
Callisto
  • Callisto
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
anonymous
  • anonymous
yes, I have and I do know the trig functions that u mentioned
Callisto
  • Callisto
Oh great. Sudden quiz. What is cos135?
anonymous
  • anonymous
in exact value or degrees?
anonymous
  • anonymous
Hint: 135 < 180
Callisto
  • Callisto
135 is in degrees. cos135 gives you an exact value.
anonymous
  • anonymous
-cos45 = -Sqrt2/2
Callisto
  • Callisto
Yes.
Callisto
  • Callisto
Can we go back to your question 2 now?
anonymous
  • anonymous
yes, please. It's late over here and I want to go to sleep lol.
anonymous
  • anonymous
but I want to figure out how to do it properly first so I can do the other ones tomorrow.
Callisto
  • Callisto
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?
anonymous
  • anonymous
wait. let me see... and good look for your quiz.
anonymous
  • anonymous
well since the value is negative it has to be in the second or third quadrant?
Callisto
  • Callisto
Yes. Another hint given by @Chlorophyll before : -1/√2 = - (1/√2) x (√2/√2) = - √2/2
anonymous
  • anonymous
from there I'm a bit stuck as to how to solve for y.
anonymous
  • anonymous
Consider y as X!
Callisto
  • Callisto
cosy = -√2/2 What is the value of angle such that cos(angle) = -√2/2? Another hint: refer to the sudden quiz
anonymous
  • anonymous
yes, but how did you get the -√2/2?
Callisto
  • Callisto
Ratinoalization?! -1/√2 = - (1/√2) x (√2/√2) = - √2/2
anonymous
  • anonymous
can't u cancel the top and bottom Sqrt 2 and so u get -1/Sqrt2
Callisto
  • Callisto
Oh yes, of course! But you need to know that cos45 = 1/√2
anonymous
  • anonymous
-√2/2 looks more familiar to you or not?
anonymous
  • anonymous
yes, it is cos 45 right?
anonymous
  • anonymous
-cos 45 in this case*
Callisto
  • Callisto
-cos45 = cos(___ -45) = cos(____+45) ?
anonymous
  • anonymous
-cos45 = cos(180-45) = cos (180+45)?
Callisto
  • Callisto
Yes. So... cosy = -1/√2 y = ... or y= ...?
anonymous
  • anonymous
wait... let me write something else down first.
Callisto
  • Callisto
Take your time~
anonymous
  • anonymous
@JayDS @Callisto giant leap improvement here =)
anonymous
  • anonymous
lol, no way, I haven't improved at all but Callisto has.
Callisto
  • Callisto
What??!!! You haven't improved at all :'( ???
anonymous
  • anonymous
√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?
anonymous
  • anonymous
I guess I have slightly lol, I'm just really stupid.
anonymous
  • anonymous
@JayDS Our goal here is to help YOU improved :)
Callisto
  • Callisto
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
  • Callisto
*delete ''you can't be stupid than me''
anonymous
  • anonymous
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.
anonymous
  • anonymous
I like*
anonymous
  • anonymous
@Callisto wanna me to take over so you have time with your quiz?
anonymous
  • anonymous
sorry for taking up your time guys and thanks for your help so far.
Callisto
  • Callisto
@Chlorophyll Ah... I forget my quiz :| I'll come back as soon as possible!!
anonymous
  • anonymous
@Callisto Good luck, Sis! Not to be worried I promise to a good job here :)
Callisto
  • Callisto
@Chlorophyll Sis, I trust you :) Thanks too!!
anonymous
  • anonymous
what's with the "Sis" lol
anonymous
  • anonymous
√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!
anonymous
  • anonymous
sorry, I'm confused about the replacing part.
anonymous
  • anonymous
Hint: the angle in the second and third quadrants
anonymous
  • anonymous
yep, I know that but... I'm unsure how to use that
anonymous
  • anonymous
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?
anonymous
  • anonymous
cos(x+90)° = -√2/2 cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)
anonymous
  • anonymous
So you have 2 cases here!
anonymous
  • anonymous
x+90° = 180-45 and .....
anonymous
  • anonymous
yeh, that is the same as what I said earlier? pi - sqrt 2/2 and pi + sqrt 2/2 I believe.
anonymous
  • anonymous
but then you would need to convert is back to degrees.
anonymous
  • anonymous
It's incorrect to connect the degree ( pi) with the radian ( arc length ) pi - sqrt 2/2
anonymous
  • anonymous
kk, can you quickly explain this line "cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)"
anonymous
  • anonymous
oh wait, I think I got it.
anonymous
  • anonymous
Don't pay attention to the left part yet!
anonymous
  • anonymous
-√2/2 = -cos45 Now we wish to get rid of negative sign: -cos45 = cos ( 180 - 45) -cos45 = cos ( 180 + 45)
anonymous
  • anonymous
That's what Callisto has been working so hard to show you the concept!
anonymous
  • anonymous
√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)
anonymous
  • anonymous
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 = ....?
anonymous
  • anonymous
x + 90 = 135???
anonymous
  • anonymous
-> x = ...?
anonymous
  • anonymous
45
anonymous
  • anonymous
Similarly for the third quadrant case, can you do it yourself?
anonymous
  • anonymous
well if x=45 wouldn't it just be 180+45=225?
anonymous
  • anonymous
cos (x+90) = cos ( 180 + 45) -> x+90 = ....?
anonymous
  • anonymous
x + 90 = 225 x = 135
anonymous
  • anonymous
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!
anonymous
  • anonymous
LOL, okay big sis ^^ I will practice tomorrow but it's getting late so I'm going to sleep now.
anonymous
  • anonymous
Okie, the solution already saved into your profile. Feel free to use it as your study note!
anonymous
  • anonymous
really, how?
anonymous
  • anonymous
is that a new function?
anonymous
  • anonymous
Click into your profile, you'll see all your questions and answers!
anonymous
  • anonymous
You can't see your own questions and answer in your profile?
anonymous
  • anonymous
Maybe you can only do because you have already reached 99 smart points or something.
anonymous
  • anonymous
SmartScore*
anonymous
  • anonymous
No, you can check anybody profiles!
anonymous
  • anonymous
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.
anonymous
  • anonymous
Well I g2g now Sis, thanks so much for the help and help me say thanks to Callisto as well.

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