Find the product. Leave the result in trigonometric form. (Enter your an \[ \left[\frac{1}{3}\left(\cos \left(140^{\circ}\right)+i \sin \left(140^{\circ}\right)\right)\right]\left[\frac{4}{5}\left(\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right)\right)\right] \]

Solve \( 8 \sin (2 \beta)+1 \cos (\beta)=0 \) for all solutions \( 0 \leq \beta<2 \pi \) \( \beta=\square \) Give your answers accurate to at least 2 decimal places and in a list separated by commas.

Prove that \( \begin{array}{ll}\text { (1) } \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x & \text { (2) } \cos 4 x=8 \cos ^{4} x-8 \cos ^{2} x+1\end{array} \)

Prove that (1) \( \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x-1 \) Prove the following identities: (1) \( \quad \frac{2+2 \cos x}{\sin 2 x}=\frac{\tan x}{1-\cos x} \) (3) \( \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} \)

Exercice 3 \( A B C \) un triangle tel que : \( A C-\sqrt{52} \mathrm{~cm} ; B H=9 \mathrm{~cm} \) et \( H C=4 \mathrm{~cm} \) Hest le profete orthogonal de \( A \) sur \( (B C) \). 1) Calculer \( A H \) et \( A B \). 2) Montrer que \( A B C \) est un triangle rectangle en \( A \). 3) Calculer \( \cos A B C ; \sin A B C \) et \( \tan A B C \)

Solve the following equation for \( \beta \) over the interval \( [0,2 \pi) \), giving exact answers in radian units. If an equation has no solution, enter DNE. Multiple solutions should be entered as a comma-separated list. \( \sin (\beta)-\cos (2 \beta)=0 \) \( \beta=\square \)