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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
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For question (1) 2 sin (x − 60)° = 1 Divide both sides by 2, what do you get?
sin(x-60)°=1/2
Yes. Next, take arcsine for both sides, what do you get?

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Other answers:

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

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