6) Solve the trigonometric equations over the interval \( 0 \leq x \leq 2 \pi \) a) \( 2 \sin 2 x-\sin x=0 \) b) \( \cos 2 x+\sin ^{2} x-4 \cos x+3=0 \) c) \( 2 \csc x^{2}-8=0 \) ( \( 2=0 \)

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

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

Solve \( 5 \cos (2 \alpha)=5 \cos ^{2}(\alpha)-4 \) for all solutions \( 0 \leq \alpha<2 \pi \) \( \alpha=\square \) Give your answers accurate to at least 2 decimal places and in a list separated by commas.

(0.50) - Qual é a medida do lado oposto ngulo de \( 30^{\circ} \), em um triângulo, sabendo que utros dois lados medem 2 e \( \sqrt{3} \) ?