1) In a right triangle with one angle measuring 40 degrees, the leg opposite the 40 degree angle is 5 cm . What is the length of the hypotenuse?

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.

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

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.

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

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