1. [SOLVED] Trig equation

$tan(2x) = \frac{1 + sin(x)}{cos(x)}$

I'm all out of ideas here. Anyone?

2. I'd start with the Right-hand side. Multiply numerator and denominator by 1 + sin(x) and see what happens.

Like many such problems, it may not lead anywhere, but it's a place to start and something else may hit you.

For example, while I was typing this, it dawned on me that I don't happen to know any formulas for tan(2x) right off the top of my head. I might be tempted to change that to sines and cosines. Maybe not.

Rule #1 on Trig Identities -- You can't break it. Just try something.

3. Thanks, but I've already tried everything I can think of. I need to see a solution soon or my head will explode.

As if the last one wasn't bad enough, I'm now stuck with this one as well:

$2cos^3(x) + sin(x) = 2cos(x)$

4. Hello, Spec!

$2\cos^3\!x + \sin x \:= \:2\cos x$

We have: . $2\cos x(\cos^2\!x) + \sin x \;=\;2\cos x \quad\Rightarrow\quad 2\cos x(1 - \sin^2\!x) + \sin x \;=\;2\cos x$

. . . . $2\cos x - 2\sin^2\!x\cos x + \sin x \;=\;2\cos x \quad\Rightarrow\quad \sin x - 2\sin^2\!x\cos x \;=\;0$

Factor: . $\sin x(1 - 2\sin x\cos x) \;=\;0$

And we have two equations to solve:

$\sin x \:=\:0\quad\Rightarrow\quad\boxed{x \:=\:\pi n}$

$1 - 2\sin x\cos x\:=\:0\quad\Rightarrow\quad 1 - \sin2x \:=\:0\quad\Rightarrow\quad \sin2x \:=\:1$
. . $2x \:=\:\frac{\pi}{2} + 2\pi n \quad\Rightarrow\quad\boxed{x \:=\:\frac{\pi}{4} + \pi n}$

5. Hello, Spec!

I'll give only the principal roots.
. . You can generalize the answers . . .

Solve for $x\!:\;\;\tan 2x \;= \;\frac{1 + \sin x}{\cos x}$

We have: . $\frac{\sin2x}{\cos2x} \;=\;\frac{1+\sin x}{\cos x} \quad\Rightarrow\quad \frac{2\sin x\cos x}{1 - 2\sin^2\!x} \;=\;\frac{1+\sin x}{\cos x}$

. . $2\sin x\cos^2x \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . $2\sin x(1 - \sin^2\!x) \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . $2\sin x - 2\sin^3\!x \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . which simplifies to: . $2\sin^2\!x + \sin x - 1 \;=\;0$

. . and factors: . $(2\sin x - 1)(\sin x + 1) \;=\;0$

And we have two equations to solve.

. . $2\sin x - 1 \:=\:0\quad\Rightarrow\quad \sin x \:=\:\frac{1}{2}\quad\Rightarrow\quad\boxed{x \:=\:\frac{\pi}{6},\:\frac{5\pi}{6}}$

. . $\sin x + 1\:=\:0\quad\Rightarrow\quad \sin x \:=\:-1\quad\Rightarrow\quad\boxed{ x \:=\:\frac{3\pi}{2}}$

6. Originally Posted by Soroban
Hello, Spec!

I'll give only the principal roots.
. . You can generalize the answers . . .

We have: . $\frac{\sin2x}{\cos2x} \;=\;\frac{1+\sin x}{\cos x} \quad\Rightarrow\quad \frac{2\sin x\cos x}{1 - 2\sin^2\!x} \;=\;\frac{1+\sin x}{\cos x}$

. . $2\sin x\cos^2x \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . $2\sin x(1 - \sin^2\!x) \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . $2\sin x - 2\sin^3\!x \;=\;1 + \sin x - 2\sin^2\!x - 2\sin^3\!x$

. . which simplifies to: . $2\sin^2\!x + \sin x - 1 \;=\;0$

. . and factors: . $(2\sin x - 1)(\sin x + 1) \;=\;0$

And we have two equations to solve.

. . $2\sin x - 1 \:=\:0\quad\Rightarrow\quad \sin x \:=\:\frac{1}{2}\quad\Rightarrow\quad\boxed{x \:=\:\frac{\pi}{6},\:\frac{5\pi}{6}}$

. . $\sin x + 1\:=\:0\quad\Rightarrow\quad \sin x \:=\:-1\quad\Rightarrow\quad\boxed{ x \:=\:\frac{3\pi}{2}}$

Thanks! $x = \frac{3\pi}{2} + 2{\pi}n$ is not a solution though, since $cos x \neq 0$.