Y = Sin(x) + Cos(x) Transpose

**ZachGriffin** · Jun15-09, 08:12 AM

Hi Guys, Simple question; I'm trying to work out the transpose of Y = Sin(x) + Cos(x) to make x the subject. I thought it would be x = arccos(arcsin(y)) / 2 however I don't think thats right. Is there another theorem I'm missing?

**Irrational** · Jun15-09, 09:01 AM

what's

?

**ZachGriffin** · Jun15-09, 09:43 AM

that is meant to be divided by 2

**Irrational** · Jun15-09, 10:03 AM

$LaTeX Code: \\Rightarrow y^2 = (sin(x) + cos(x))^2$

$LaTeX Code: \\Rightarrow y^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x)$

$LaTeX Code: \\Rightarrow y^2 = \\left(sin^2(x) + cos^2(x)\\right) + 2sin(x)cos(x)$

$LaTeX Code: \\Rightarrow y^2 = 1 + 2sin(x)cos(x)$

$LaTeX Code: \\Rightarrow y^2 - 1 = 2sin(x)cos(x)$

$LaTeX Code: \\Rightarrow y^2 - 1 = sin(2x)$

$LaTeX Code: \\Rightarrow 2x = \\sin^{-1 }(y^2 - 1)$

$LaTeX Code: \\Rightarrow x = \\frac { \\sin^{-1 } (y^2 - 1) } {2}$

**ZachGriffin** · Jun15-09, 10:36 AM

Thanks very much for that. Anyone looking for the rules for this, I've found them on http://math2.org/math/trig/identities.htm

sin(2x) = 2 sin x cos x

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

cos^2(x) = 1/2 + 1/2 cos(2x)

**Irrational** · Jun15-09, 10:38 AM

should be one of the first identities you learn.

**Mute** · Jun15-09, 11:05 AM

You can put it into another form by noting:

$LaTeX Code: \\sin x + \\cos x = \\sqrt{2}\\left[\\sin x \\left(\\frac{1}{\\sqrt{2}}\\right) + \\cos x \\left(\\frac{1}{\\sqrt{2}} \\right)\\right] = \\sqrt{2}\\left[\\sin x \\cos \\frac{\\pi}{4} + \\cos x \\sin \\frac{\\pi}{4} \\right] = \\sqrt{2}\\sin \\left(x+\\frac{\\pi}{4}\\right)$ .

So, y is also equal to this expression and you can solve for x. Since you don't seem to be familiar with many trig identities, I used the sine "angle sum formula", $LaTeX Code: \\sin(a \\pm b) = \\sin a \\cos b \\pm \\cos a \\sin b$ .

**ZachGriffin** · Aug8-09, 06:09 AM

Suppose I was to change the equation to Y = a sin(x) + b cos(x). How would I transpose it to make x the subject? I've searched through identities on that site but can't find one that relates to having 2 different coefficients a and b.

**Hurkyl** · Aug8-09, 06:39 AM

Originally Posted by ZachGriffin

Suppose I was to change the equation to Y = a sin(x) + b cos(x). How would I transpose it to make x the subject? I've searched through identities on that site but can't find one that relates to having 2 different coefficients a and b.

Think hard about how Mute's example works....

**ZachGriffin** · Aug8-09, 06:57 AM

A bit more searching I've come across this:

a sin x + b cos x = R sin (x + alpha) which is an R Formulae with R = Sqrt(a^2 + b^2) and alpha = atan(b / a). If I use that I should be able to solve for x.

Having a look at Mute's post, what should I be looking for? It's been a few years since I left school so most of this is going back to that. I'd rather know how this thing works than just the answer so I'll keep looking.

**Hurkyl** · Aug8-09, 07:17 AM

Originally Posted by ZachGriffin

Having a look at Mute's post, what should I be looking for?

To derive the formula you just found by searching -- his calculation is the derivation of that formula, just in a special case.