Question

find the tangent of $\angle y$. simplify your answer and write it as a proper fraction, improper fraction, or whole number. $\tan (y) = \square$

Explanation:

Step1: Recall the tangent ratio in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For \(\angle Y\) in right triangle \(XYZ\) (right - angled at \(X\)):

• The side opposite to \(\angle Y\) is \(XZ\).
• The side adjacent to \(\angle Y\) is \(XY\).

First, we need to find the length of \(XZ\) using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\), \(b\) are the legs. Here, \(YZ = 73\) (hypotenuse), \(XY = 48\) (one leg), and \(XZ\) is the other leg. So, \(XZ=\sqrt{YZ^{2}-XY^{2}}\)
\[XZ=\sqrt{73^{2}-48^{2}}=\sqrt{(73 + 48)(73 - 48)}=\sqrt{121\times25}=\sqrt{121}\times\sqrt{25}=11\times5 = 55\]

Step2: Calculate \(\tan(Y)\)

Now, \(\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}=\frac{XZ}{XY}\)
We know that \(XZ = 55\) and \(XY=48\), so \(\tan(Y)=\frac{55}{48}\)

Answer:

\(\frac{55}{48}\)