User:Tikai/math/q

$$\begin{align} \sin x +\cos 2x = \sin x+ \cos^2 x -\sin^2 x \\ =1-2\sin^2 x +\sin x \\ =1-2\left (\sin x-\frac{1}{4}\right )^2+\frac{1}{8}\\ =\frac{9}{8}-2\left (\sin x-\frac{1}{4}\right )^2\\ \end{align}$$

$$ \begin{cases} if x=\frac{1}{4} max=\frac{9}{8}\\ if x=-1 min=\frac{9}{8}-2\times \frac{25}{16}\\ =-2 \end{cases} $$

$$\begin{cases} \csc 10 \csc 30 \csc 50 \csc 70 \\ = \frac{1}{\sin 10 \sin 30 \sin 50 \sin 70} \\ = \cfrac{2}{\cos 80 \cos 40 \cos 20} = k \\ \frac{1}{8\sin 20}k = \frac{2}{\sin 160} = \frac{2}{\sin 20} \\ k = 2\times 8 = 16\\ = \cfrac{2}{\sin 10 \sin 50 \sin 70} \\ = \cfrac{2}{\sin 10 \sin (60-10) \sin (60+10)} \\ = \cfrac{2}{\frac{1}{4}\sin (3 \times 10)} \end{cases}$$

$$\begin{align} \sin 20 \sin 40 \sin 80 \\ & = \sin 80 \sin 40 \sin 20 \\ & = \frac{\cos (80-40) - \cos (80+40)}{2} \sin 20 = \frac{1}{2}(\cos 40 - \cos 120) \sin 20 \\ & = \frac{1}{2} (\sin20 \cos 40 + \sin 20 \frac{1}{2}) \\ & = \frac{\sin 20}{4} + \frac{1}{2} \sin20 \cos 40 \\ & = \frac{\sin 20}{4} + \frac{1}{4}(\sin 60-\sin 20) \\ & = \frac{\sin 60}{4} = \frac{\sqrt{3}}{8} \end{align}$$

$$\begin{align} \sin 20 + \sin 40 - \sin 80 \\ & = -(\sin 80- \sin 40 - \sin 20) \\ & = -(2\cos (\cfrac{80+40}{2}) \sin (\cfrac{80-20}{2}) - \sin 20) \\ & = -(2(\cos 60\sin 20))+ \sin 20 \\ & = 0 \end{align}$$

7
$$\begin{align} \sin ^2 2t -3 \cos ^2 t \\ & = \left( 2\sin t\cos t  \right)^2 - 3\cos ^2 t \\ & = \cos ^2 t \left( 4-4\cos^2 t -3 \right) \\ & = \cos ^2 t-4\cos ^4 t \\ & = -4\left( \cos ^2 t - \frac{1}{2} \right)^2 + \frac{1}{8} \end{align} \begin{cases} \cos t = \frac{1}{2}, & \frac{1}{8} \\ \cos t = -1, & \frac{-71}{8} \end{cases}$$

6
$$\begin{align} 0<\alpha ,\beta < 90 \\ \sin \alpha + \sin \beta = a \\ \cos \alpha + \cos \beta = b \\ & a^2 = \sin ^2 \alpha + \sin ^2 \beta + 2\sin \alpha \sin \beta \\ & b^2 = \cos ^2 \alpha + \cos ^2 \beta + 2\cos \alpha \cos \beta \\ b^2 + a^2 = 2 + \cos \alpha -\beta \Rightarrow wrong \\ & a = 2 \sin \frac{\alpha +\beta}{2}\cos \frac{\alpha -\beta}{2} \\ & b = 2 \cos \frac{\alpha +\beta}{2}\cos \frac{\alpha -\beta}{2} \\ \cfrac{a}{b} = \tan \frac{\alpha +\beta}{2} \\ \cos 2\theta = \cfrac{\tan ^2 \theta - 1}{\tan ^2 \theta + 1} \\ \sin 2\theta = \cfrac{2\tan \theta}{\tan ^2 \theta + 1} \\ \cos \alpha +\beta = \cfrac{\cfrac{a}{b}^2 - 1}{\cfrac{a}{b}^2 + 1} \\ \sin \alpha +\beta = \cfrac{2\times \cfrac{a}{b}^2}{\cfrac{a}{b}^2 + 1} \\ \end{align}$$