If \( \theta=-\frac{4 \pi}{3} \), then find exact values for the following: \[ \begin{aligned} \sin (\theta) & =\square \\ \cos (\theta) & =\frac{-1}{2} \\ \tan (\theta) & =-\sqrt{3} \\ \csc (\theta) & =\square\end{aligned} \] \( \begin{aligned} \sec (\theta) & =\square\end{aligned} \)

Solve \( 6 \cos (2 t)=6 \cos ^{2}(t)-2 \) for all solutions \( 0 \leq t<2 \pi \). \( t=\square \) Give your answers accurate to at least 2 decimal places and in a list separated by commas.

Solve \( 4 \sin \left(\frac{\pi}{4} x\right)=2 \) for the four smallest positive solutions. \( x=\square \) Give your answers accurate to at least two decimal places, as a list separated by commas.

Solve \( 9 \cos (2 \alpha)=9 \sin ^{2}(\alpha)+4 \) 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.

Using the Law of Sines to solve for all possible triangles if \( \angle B=43^{\circ}, a=109, b=42 \). If no answer exists, enter DNE for all answers. \( \angle A \) is \( \square \) degrees \( \angle C \) is \( \square \) degrees \( c=\square \) Assume \( \angle A \) is opposite side \( a, \angle B \) is opposite side \( b \), and \( \angle C \) is opposite side \( c \).

Using the Law of Sines to solve the triangle if \( \angle A=37^{\circ}, \angle C=72^{\circ}, b=12: \) \( \angle B \) is 71 \( a=7.63 \) \( c=\square \) degrees; Round to two decimal places if needed. Assume \( \angle A \) is opposite side \( a, \angle B \) is opposite side \( b \), and \( \angle C \) is opposite side \( c \).