Q. Calcular el ángulo de los siguientes númeios complejos a) \( 3+2 i \) b) \( \frac{5}{2}+\frac{3}{7} i \)

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

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. \( \tan ^{2}(\beta)-1=-2 \tan (\beta) \) \( \beta=\square \)

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) \( \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} \)

El producto escalar o producto punto entre dos vectores \( \vec{u} \) y \( \vec{v} \) se define como \( \vec{u} \cdot \vec{v}=|\vec{u}||\vec{v}| \cos \theta \), donde \( \theta \) es el ángulo entre ambos vectores. Si \( \vec{u} \) es ortogonal a \( \vec{v} \), entonces sobre el ángulo \( \theta \) es ciertc que

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. \( \sin (\beta)-\cos (2 \beta)=0 \) \( \beta=\square \)