Simplify \( \csc \left(\frac{\pi}{2}\right) \)

5. Si \( \operatorname{sen} \theta=\frac{3}{5} \) determina el valor de \( M=3 \operatorname{tg} \theta \) \( \begin{array}{lll}\text { a. } \frac{3}{4} & \text { b. } 7 \frac{3}{4} & \text { c. } \frac{9}{4} \\ \text { d. } 1 \frac{3}{4} & \text { e. } \frac{35}{4} & \end{array} \)

For what values of the angle between \( \vec{A} \) and \( \vec{B} \) will the dot product \( \vec{A} \vec{B} \) be positive, negative or zero?

\( \left. \begin{array} { l }{ 40 ^ { \circ } 12 ^ { \prime } 48 ^ { \prime \prime } } \\ { + 17 ^ { \circ } \quad 5 ^ { \prime } 59 ^ { \prime \prime } } \end{array} \right. \)

3. Prove \( \sin 3 x=3 \cos ^{2} x \sin x-\sin ^{2} x \) 4. Prove \( \sec 2 x=\frac{\csc ^{2} x}{\csc ^{2} x-2} \) 5. Prove \( \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=2 \csc \theta \) 6. Prove \( \frac{c s c^{2} \theta-1}{c s^{2} \theta}=\cos ^{2} \theta \) 7. Prove \( \frac{1}{1-\cos ^{2}}+\frac{1}{1+\cos \mathrm{B}}=2 \csc ^{2} \) 8. Prove \( \left(\cot ^{2} A+1\right)\left(\sin ^{2} A-1\right)=-\cot ^{2} A \)

12. Jordan is standing on top of a 10 m diving platform. His swimming coach is on the ground to the left of him, while his parents are on the ground to the right of him. His coach is looking at him at an angle of \( 37^{\circ} \) with the ground, while his parents are looking at him at an angle of \( 32^{\circ} \) with the ground. Jordan rotates his head \( 100^{\circ} \) left to right to see his coach and parents. How far apart are his coach and parents? Round to one decimal place.