Question

express tan n as a fraction in simplest terms.

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 \( N \) in right triangle \( LMN \) (right - angled at \( M \)):
\(\tan N=\frac{\text{opposite side to }N}{\text{adjacent side to }N}\)
The side opposite to \( N \) is \( LM=\sqrt{71}\), and we need to find the length of the adjacent side \( MN \).

Step2: Use the Pythagorean theorem to find \( MN \)

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 \( LN = \sqrt{96}\), leg \( LM=\sqrt{71}\), and leg \( MN=x \) (let's say).
So, \( LM^{2}+MN^{2}=LN^{2}\)
\((\sqrt{71})^{2}+x^{2}=(\sqrt{96})^{2}\)
\(71 + x^{2}=96\)
Subtract 71 from both sides: \(x^{2}=96 - 71=25\)
Take the square root of both sides: \(x = 5\) (since length is positive)

Step3: Calculate \(\tan N\)

Now that we know the opposite side (\( LM=\sqrt{71}\)) and the adjacent side (\( MN = 5\)) to angle \( N \), using the definition of tangent:
\(\tan N=\frac{LM}{MN}=\frac{\sqrt{71}}{5}\)

Answer:

\(\frac{\sqrt{71}}{5}\)