4sin(x)cos(x) + 1 = 2(sin(x) + cos(x)); need help understanding the solutions

[Captain Sunshine]

I've just completed the following problem from a trig book I'm working through, however I don't understand why the solutions given are correct. I was hoping someone could provide a brief explanation as to why these solutions are the right ones.

Here's the problem:
Solve for (x), giving all solutions in the interval -180 to 180. (The solutions given are: -60, 30, 60, 150).


4sin(x)cos(x) + 1 = 2(sin(x) + cos(x))


My working is:

16 sin(x)^2 cos(x)^2 + 1 = 4(sin(x)^2 + cos(x)^2) (square)

16 sin(x)^2 (1 - sin(x)^2) + 1 = 4(sin(x)^2 + (1 - sin(x)^2)) (use trig identity to replace cos)

16(sin(x)^2 - sin(x)^4) + 1 = 4sin(x)^2 + 4 - 4sin(x)^2 (expand)

16sin(x)^2 - 16sin(x)^4 - 3 = 0 (subtract 4sin(x)^2 + 4 - 4sin(x)^2)

(4sin(x)^2 - 1) (4sin(x)^2 - 3) (factor the quadratic)

sin(x)^2 = 1/4 or 3/4

sin(x) = ± 1/2 or ± √3/2 (square root)


Okay, so here's where I hit a brick wall. When I evaluate these I get:

arcsin (1/2) = 30, with its equivalent angle in quadrant 2 being 150. So far so good.

arcsin (-1/2) = -30 (this was not a given solution, nor -150).

arcsin (√3/2) = 60, which is good, but 120 isn't a solution.

arcsin (-√3/2) = -60 (again, -120 isn't a solution).


I can't figure out why only the principal root of one solution (and its equivalent angle) is accepted, but both roots of the other solution are fine (but their equivalent angles are not!). I can only assume I've overlooked some salient information, or my working is faulty.

As a little more background, thus far the book I'm working through (Understand Trigonometry. Ironically having the opposite effect at the moment) has not covered anything more complex than the basic trig identities: sin^2(x) + cos^2(x) = 1, tan^2(x) + 1 = sec^2(x), 1 + cot^2(x) = cosec^2(x).
I'm operating under the assumption that this should be sufficient to solve the problem.

Thanks in advance for any help or advice given; it's all appreciated.

 

[Deleted member 4993]

I've just completed the following problem from a trig book I'm working through, however I don't understand why the solutions given are correct. I was hoping someone could provide a brief explanation as to why these solutions are the right ones.

Here's the problem:
Solve for (x), giving all solutions in the interval -180 to 180. (The solutions given are: -60, 30, 60, 150).


4sin(x)cos(x) + 1 = 2(sin(x) + cos(x))


My working is:

16 sin(x)^2 cos(x)^2 + 1 = 4(sin(x)^2 + cos(x)^2) (square) ..... How did you get that ! Remember (a+b)² = a² + b² + 2*a*b
16 sin(x)^2 (1 - sin(x)^2) + 1 = 4(sin(x)^2 + (1 - sin(x)^2)) (use trig identity to replace cos)

16(sin(x)^2 - sin(x)^4) + 1 = 4sin(x)^2 + 4 - 4sin(x)^2 (expand)

16sin(x)^2 - 16sin(x)^4 - 3 = 0 (subtract 4sin(x)^2 + 4 - 4sin(x)^2)

(4sin(x)^2 - 1) (4sin(x)^2 - 3) (factor the quadratic)

sin(x)^2 = 1/4 or 3/4

sin(x) = ± 1/2 or ± √3/2 (square root)


Okay, so here's where I hit a brick wall. When I evaluate these I get:

arcsin (1/2) = 30, with its equivalent angle in quadrant 2 being 150. So far so good.

arcsin (-1/2) = -30 (this was not a given solution, nor -150).

arcsin (√3/2) = 60, which is good, but 120 isn't a solution.

arcsin (-√3/2) = -60 (again, -120 isn't a solution).


I can't figure out why only the principal root of one solution (and its equivalent angle) is accepted, but both roots of the other solution are fine (but their equivalent angles are not!). I can only assume I've overlooked some salient information, or my working is faulty.

As a little more background, thus far the book I'm working through (Understand Trigonometry. Ironically having the opposite effect at the moment) has not covered anything more complex than the basic trig identities: sin^2(x) + cos^2(x) = 1, tan^2(x) + 1 = sec^2(x), 1 + cot^2(x) = cosec^2(x).
I'm operating under the assumption that this should be sufficient to solve the problem.

Thanks in advance for any help or advice given; it's all appreciated.

.

 

[Captain Sunshine]

You're right! What a stupid error! Okay, after you very helpfully pointed that out, I redid my working:

16sin²(x) cos²(x) + 8sin(x)cos(x) + 1 = 4(sin²(x) + 2sin(x)cos(x) +cos²(x)) (this is what I believe is correct now)

16sin²(x) cos²(x) + 8sin(x)cos(x) + 1 = 4sin²(x) + 8sin(x)cos(x) +4cos²(x) (expanded)

16sin²(x) cos²(x) + 1 = 4sin²(x) + 4cos²(x) (subtract 8sin(x)cos(x) from both sides)

16 sin²(x) (1 - sin²(x)) + 1 = 4sin²(x) + 4(1 - sin²(x)) (Replace cos with sin using trig identity)

16sin²(x) - 16sin⁴(x) + 1 = 4sin²(x) + 4 - 4sin²(x)

16sin²(x) - 16sin⁴(x) + 1 = 4
- 16sin⁴(x) + 16sin²(x) - 3 = 0

16sin⁴(x) - 16sin²(x) + 3 = 0 (multiplied by (-1), just because I find it easier to factor)

From here, the working is identical to my previous post.


Basically, it looks like I skipped a few steps when copying out my working (again apologies for the silly oversight), but the end result is the same. Thank you for taking the time out to look at this for me, I am very grateful. This is the second day I've been looking at this, but I'm still not able to understand the solutions.

 

[Ishuda]

I've just completed the following problem from a trig book I'm working through, however I don't understand why the solutions given are correct. I was hoping someone could provide a brief explanation as to why these solutions are the right ones.

Here's the problem:
Solve for (x), giving all solutions in the interval -180 to 180. (The solutions given are: -60, 30, 60, 150).


4sin(x)cos(x) + 1 = 2(sin(x) + cos(x))


My working is:

16 sin(x)^2 cos(x)^2 + 1 = 4(sin(x)^2 + cos(x)^2) (square)

16 sin(x)^2 (1 - sin(x)^2) + 1 = 4(sin(x)^2 + (1 - sin(x)^2)) (use trig identity to replace cos)

16(sin(x)^2 - sin(x)^4) + 1 = 4sin(x)^2 + 4 - 4sin(x)^2 (expand)

16sin(x)^2 - 16sin(x)^4 - 3 = 0 (subtract 4sin(x)^2 + 4 - 4sin(x)^2)

(4sin(x)^2 - 1) (4sin(x)^2 - 3) (factor the quadratic)

sin(x)^2 = 1/4 or 3/4

sin(x) = ± 1/2 or ± √3/2 (square root)


Okay, so here's where I hit a brick wall. When I evaluate these I get:

arcsin (1/2) = 30, with its equivalent angle in quadrant 2 being 150. So far so good.

arcsin (-1/2) = -30 (this was not a given solution, nor -150).

arcsin (√3/2) = 60, which is good, but 120 isn't a solution.

arcsin (-√3/2) = -60 (again, -120 isn't a solution).


I can't figure out why only the principal root of one solution (and its equivalent angle) is accepted, but both roots of the other solution are fine (but their equivalent angles are not!). I can only assume I've overlooked some salient information, or my working is faulty.

As a little more background, thus far the book I'm working through (Understand Trigonometry. Ironically having the opposite effect at the moment) has not covered anything more complex than the basic trig identities: sin^2(x) + cos^2(x) = 1, tan^2(x) + 1 = sec^2(x), 1 + cot^2(x) = cosec^2(x).
I'm operating under the assumption that this should be sufficient to solve the problem.

Thanks in advance for any help or advice given; it's all appreciated.

Besides the mistake poined out above, before I start to square sides or look at 'simplifications' like 2sin(x)cos(x)=sin(2x), I consider whether I can get a cosine (or sine) by itself with a 'nice' remaining part. Looking at this equation, if we rearrange it we have
2 (2 sin(x) - 1) cos(x) = 2 sin(x) - 1
which gives us two situations to work with
(a) 2 sin(x) - 1 = 0
(b) 2 sin(x) - 1 \(\displaystyle \ne\) 0.
Can you continue from there?


BTW, note that if you interchange the sine and cosine you still have the same equation:
4cos(x)sin(x) + 1 = 2(cos(x) + sin(x))
So, if you get to a solution sin(x)=a, then another solution is cos(x)=a.

 

[Captain Sunshine]

Ishuda, thank you; it hadn't occurred to me that the equality could be solved for both sine and cosine. That is very helpful, and clears up part of my confusion.

I don't understand your other point, though. You simplified it to this:

2sin(x)cos(x)=sin(2x)

And rearranged it to this:

2 (2 sin(x) - 1) cos(x) = 2 sin(x) - 1


I must admit, I have no idea how you got to this.
With the textbook I'm working through, I've not seen anything resembling this. I'm beginning to suspect that I have not yet learnt the tools needed for a problem like this, and perhaps it is covered in a later chapter. Does the technique/method you used here have a name? Or is there a link you could provide me, so that I might fill in the gaps in my knowledge. Obviously, I don't wish to waste your time with questions I could figure out for myself.

Thank you for your patience, and apologies if I'm being a little slow.