I have equation which I can not understand until end, so your help I would appreciate. I start with equation No. 1 which is clear: The relations between the phase quantities $ \underline G_r,\underline G_s,\underline G_t $ and symmetrical components $\underline G_0,\underline G_1,\underline G_2 $ are given by $$ \begin{bmatrix} \underline G_0\\ \underline G_1\\ \underline G_2\\ \end{bmatrix} = 1/3 \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \\ \end{bmatrix} \begin{bmatrix} \underline G_r\\ \underline G_s\\ \underline G_t\\ \end{bmatrix} \tag{1} $$ where $a = e^{j\frac{2\pi}3}$ and $a^2 = e^{j\frac{4\pi}3}$ are phase rotating operators.
Equation No.2 which is clear as well: The time functions of the phase quantities $q_r,q_s,q_t$
$$g_r = \widehat g_r \cos(\omega t + \varphi_r )\\
g_s = \widehat g_s \cos(\omega t + \varphi_s )\\
g_t = \widehat g_t \cos(\omega t + \varphi_t ) \tag{2}
$$
Equation No. 3: corresponding complex phasors are given by
$$ \underline G_r = \widehat g_r e^{j\varphi_r} \\
\underline G_s = \widehat g_s e^{j\varphi_s} \\
\underline G_t = \widehat g_t e^{j\varphi_t} \tag{3}
$$
From equations (2) and (3) it follows that the time function can be found according to: Equation No. 4 $$ g_i(t) = Re[\underline G_i e^{j\omega t}] = \frac{1}{2}[\underline G_i e^{j\omega t} + \underline G_i^* e^{-j\omega t} ]\tag{4}$$
Applying expressions (3) to (1) and using equation (4) it can be found equation No. 5 which I can not get: $$ 3 g_1 = \widehat g_r \cos(\omega t + \varphi_r ) -\frac{1}{2}[\widehat g_s \cos(\omega t + \varphi_s ) + \widehat g_t \cos(\omega t + \varphi_t )] -\frac{\sqrt {3} }{2}[\widehat g_s \sin(\omega t + \varphi_s ) - \widehat g_t \sin(\omega t + \varphi_t ] \tag{5}$$
I can not get this Equation No.5. I don't understand from where these $\sin$ functions came?
Note that $$3\underline G_1 = \underline G_r + a\underline G_s + a^2\underline G_t = \widehat q_r e^{\varphi_r} + \widehat q_s e^{j(\varphi_s+2\pi/3)} + \widehat q_t e^{j(\varphi_t-2\pi/3)}.$$ Consequently, \begin{align} 3\Re{\left[\underline G_1 e^{j w t}\right]} & = \Re{\left[\widehat q_r e^{\varphi_r+wt} + \widehat q_s e^{j(\varphi_s+wt+2\pi/3)} + \widehat q_t e^{j(\varphi_t+wt-2\pi/3)}\right]} \\ & = \widehat q_r \cos(wt + \varphi_r) + \widehat q_s \cos(wt + \varphi_s +2\pi/3) + \widehat q_t \cos(wt + \varphi_t-2\pi/3). \end{align} Now use $$\cos(\theta \pm a) = \cos(\theta)\cos(a) \mp \sin(\theta) \sin(a)$$ to get the final result.