Question

find the measure of angle z? round to the nearest hundredth. 42.34° .0058° 47.66° 36.47°

Explanation:

Step1: Apply the Law of Cosines

Let the sides of the triangle be \(a = 17\), \(b = 23\), and \(c\) be the side opposite angle \(Z\). The Law of Cosines formula for finding an angle \(\theta\) is \(\cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). In this case, if we assume the side - opposite angle \(Z\) is \(c\), and we want to find \(\cos Z=\frac{17^{2}+23^{2}-c^{2}}{2\times17\times23}\). But we can also use the Law of Sines. The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Let's assume the side opposite angle \(Z\) is \(c\), and we know two sides \(a = 17\) and \(b = 23\). We use \(\sin Z=\frac{a\sin B}{b}\). First, we assume the triangle is non - right - angled. Using the Law of Sines: \(\sin Z=\frac{17\sin B}{23}\). But if we assume the triangle is a right - angled triangle (not given, but for simplicity, we can also use the inverse - sine function for a right - angled triangle situation), \(\sin Z=\frac{17}{23}\) (assuming 17 is the opposite side and 23 is the hypotenuse).

Step2: Calculate the angle

\(Z=\sin^{- 1}(\frac{17}{23})\)
\(Z=\sin^{-1}(0.73913)\)
\(Z\approx47.66^{\circ}\)

Answer:

C. \(47.66^{\circ}\)