'Further trigonometric equations' - Need help understanding question procedure

Inertia_Squared

Inertia_Squared

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Hey there, I've been working on some trig questions as revision for Y11 ext1/introduction to new Y12 content, but I've been having major issues, I'm not sure how or why but for some reason I just have no clue how to do a major part in the final steps of these sorts of questions of finding multiple angles within a restricted domain. I'm fairly certain I was taught it before but for some reason, it did not stick at all, and I'm getting a little desperate/worried as it is really just fundamental stuff.

I'll have some example questions and my general thought process listed below, including where I'm having problems, and I was wondering if I would be able to have the area I'm having some trouble with explained to me in a bit of a step-by-step process of the concept itself so I can really take it apart in my head - or even a pointer towards some online resources (I tried searching online first but I wasn't able to really find what I was looking for) would be great. Thanks in advance for the help, and even if you don't help, thanks for at least taking your time to read up to this point!

So,
a good example question where this is giving me problems is in a question as such;
Solve for [0,360], sinx*cosx = 1/2*sqrt2​
For convenience, the answer stated in the textbook solution is: x = 22.5deg, 67.5deg, 202.5deg, 247.5deg​
​
My process:​
First thing I realise is that it is very close to a double angle identity, so multiplying by two we get:​
2sinx*cosx = 1/sqrt2 = sin2x​
now we know from our exact values that 1/sqrt2 = 45 with both sin and cos, but for the sake of thoroughness;​
asin(1/sqrt2) = asin(sin2x)​
=asin(1/sqrt2) = 2x => 45 deg​
​
so, from this I would assume that the solution for x is 22.5 degrees, and yes I am aware that I must minus and plus from certain critical points to gain multiple solutions within the restricted domain, but this is the area I'm having trouble with, so much so that I'm not even going to try and find further solutions, as I am so lost that I literally have no clue how to go about this in an intuitive manner. If you told me that you just minused and added from such and such points I'd say 'fair enough, kinda makes sense', but as I am now I would not be able to replicate it at all just by seeing the solution itself or even some worked solutions.​
​
I'm fully aware of the importance of factors like every 90deg interval and I'm aware of the quadrant pos/neg values for sin cos and tan, but I just cannot form an intuitive understanding of how to apply that knowledge in my head whatsoever, so any help to better this understanding is greatly appreciated.
​
Thanks again, and if you have any further questions on prerequisite knowledge, example questions or anything at all I'll do my best to answer them!
​
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Ok let's see if I can help.
So you have sin(2x) = 1/sqrt2.
And you know that sin 45 = 1/sqrt2. That's all good. Let's take a step back before we worry about the 2x.
If you had sin A = 1/sqrt 2 and A was between O and 360, what would A be?
 
Harry_the_cat said:
Ok let's see if I can help.
So you have sin(2x) = 1/sqrt2.
And you know that sin 45 = 1/sqrt2. That's all good. Let's take a step back before we worry about the 2x.
If you had sin A = 1/sqrt 2 and A was between O and 360, what would A be?
Click to expand...
Given the ambiguous case, it should be either 45 or 135 if I'm not mistaken?
 
Yes correct. Now back to sin (2x) =1/sqrt2.
If x is between 0 and 360, what must 2x be between?
 
Yes thats better!
So, if sin (2x)= 1/sqrt2, then what could 2x be? Hint: there are 4 solutions
 
damn I should really know this but I'm super tired atm, can I get a lifeline on this one? Sorry this is a little embarassing but my brain is a little all over the place atm.

For the sake of giving it a go, could it be that the solutions are 45, 135, 2pi+45 and 135 + 2pi and then all values divided by 2, i.e. 22.5, 67.5...?
 
Don't be embarassed, asking questions and giving it a go is the best way to learn.
First of all, don't mix radians and degrees. Work with one or the other. Because your original question said [0,360], it implies you are working in degrees.
So here 2x could be 45, 135, 360+45=405 or 360+135=495.
And then divide all by 2 to get back to x.
 
oh wow that makes a lot more sense than what I expected XD

this sounds cliche but its all coming back to me now, thanks!
Just to be sure I've got a handle on things could I run one more slightly harder question by you?
 
Cheers!
The question goes:
Solve for [0,2pi], sin(x-pi/4) - sin(x+pi/4) = 0.25
we'll let pi/4 = p for brevity,
I know how to do the angle sums, sinxcosp - cosxsinp - sinxcosp - cosxsinp
which goes to: 2cosxsinp = -0.25
I was gonna go a little further but my brain is breaking a little so I might get your help on this bit too so I'm not sitting here staring at a screen for 10 more minutes lol, what should I do from here?
 
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