8. ( 9 points) An architectural team is designing a stairway for an office building. The steep angle of the main stairway is designed to form an angle \( \theta \) with the ground. To create a more accessible stairway for visitors, they plan to design an alternate stairway with half the steepness of the main one. If the steep angle of the main stairway is such that \( \tan (\theta)=7 / 4 \), what is the measurement of the angle for the more accessible stairway?

5. Se observa un semáforo desde la ventana de un edificio, a 30 m del suelo. El ángulo de depresión para observar la parte superior del semáforo es de \( 70^{\circ} \), mientras que el ángulo de depresión para observar su base mide \( 88^{\circ} \). ¿Cuál es la altura del semáforo?

Which transformations are needed to change the parent cosine function to \( y=3 \cos (10(x-\pi)) \) ? vertical compression of 3 , horizontal stretch to a period of \( 5 \pi \), phase shift of \( \pi \) units to the left vertical stretch of 3 , horizontal compression to a period of \( \frac{\pi}{5} \), phase shift of \( \pi \) units to the right vertical compression of 3 , horizontal stretch to a period of 10 p, phase shift of \( \pi \) units to the right vertrech of 3 , horizontal compression to a period of \( \frac{\pi}{5} \), phase shift of \( \pi \) units to the left

What is the period of \( y=\sin (3 x) \) ? \( \begin{array}{l}\frac{\pi}{3} \\ \frac{2 \pi}{3} \\ 3 \pi \\ 6 \pi\end{array} \)

Prove the following trigonometric identity: \( \frac{\sin A\left[1-\sin \left(360^{\circ}-A\right)\right]}{1-\cos ^{2}\left(180^{\circ}+A\right)}-1=\operatorname{cosec} A \)

A buoy starts at a height of 0 in relation to sea level and then goes up. Its maximum displacement in either direction is 6 feet, and the time it takes to go from its highest point to its lowest point is 4 seconds. Which of the following equations can be used to model \( h \), the height in feet of the buoy in relation to sea level as a function of time, \( t \), in seconds? \( h=4 \sin \left(\frac{\pi}{6} t\right) \) \( h=4 \sin \left(\frac{\pi}{3} t\right) \) \( h=6 \sin \left(\frac{\pi}{4} t\right) \) \( h=6 \sin \left(\frac{\pi}{2} t\right) \)