Question

sin z =
tan z =
cos z =

Explanation:

Step1: Recall trigonometric - ratio definitions

In right - triangle \(ABC\) with right - angle at \(C\), for an acute angle \(Z\) (assuming \(\triangle ABC\) and \(\triangle XYZ\) are similar and angle \(Z\) corresponds to angle \(B\)), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Calculate \(\sin Z\)

If we consider the angle \(Z\) corresponding to angle \(B\) in \(\triangle ABC\), the opposite side to angle \(B\) is \(AC = 33\) and the hypotenuse \(AB=43.8\). So, \(\sin Z=\frac{33}{43.8}=\frac{330}{438}=\frac{55}{73}\approx0.753\).

Step3: Calculate \(\tan Z\)

The opposite side to angle \(Z\) (corresponding to \(B\)) is \(AC = 33\) and the adjacent side is \(BC = 28.8\). So, \(\tan Z=\frac{33}{28.8}=\frac{330}{288}=\frac{55}{48}\approx1.146\).

Step4: Calculate \(\cos Z\)

The adjacent side to angle \(Z\) (corresponding to \(B\)) is \(BC = 28.8\) and the hypotenuse \(AB = 43.8\). So, \(\cos Z=\frac{28.8}{43.8}=\frac{288}{438}=\frac{48}{73}\approx0.658\).

Answer:

\(\sin Z=\frac{55}{73}\), \(\tan Z=\frac{55}{48}\), \(\cos Z=\frac{48}{73}\)