Question

find: \tan z

Explanation:

Step1: Recall the definition of tangent 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 \( Z \) in right triangle \( XYZ \) (right - angled at \( X \)):
\(\tan Z=\frac{\text{opposite side to }Z}{\text{adjacent side to }Z}\)
The side opposite to angle \( Z \) is \( XY \) with length \( 60 \), and the side adjacent to angle \( Z \) is \( XZ \). 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, hypotenuse \( YZ = 87 \) and one leg \( XY=60 \). Let \( XZ = x \), then \( 60^{2}+x^{2}=87^{2} \).
\(x^{2}=87^{2}-60^{2}\)
We know that \( a^{2}-b^{2}=(a + b)(a - b) \), so \( 87^{2}-60^{2}=(87 + 60)(87 - 60)=(147)(27)=3969\)
Then \( x=\sqrt{3969} = 63\)

Step2: Calculate \(\tan Z\)

Now that we know the length of the opposite side (\( XY = 60 \)) and the adjacent side (\( XZ=63 \)) to angle \( Z \), using the definition of tangent:
\(\tan Z=\frac{XY}{XZ}=\frac{60}{63}=\frac{20}{21}\)

Answer:

\(\frac{20}{21}\)