Question

given right triangle abc, what is the value of tan(a)?
$\frac{5}{13}$
$\frac{12}{13}$
$\frac{12}{5}$
$\frac{13}{12}$

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For angle \( A \), we need to identify the opposite and adjacent sides.

Step2: Identify sides relative to angle \( A \)

Assuming the right triangle \( ABC \) with the right angle at \( C \) (a common convention if not specified otherwise), for angle \( A \):

• The opposite side to angle \( A \) would be \( BC \).
• The adjacent side to angle \( A \) would be \( AC \).

From the common Pythagorean triple (5, 12, 13), if the sides are 5, 12, 13 (where 13 is the hypotenuse), then:

• Opposite side to \( A \): 12 (assuming \( BC = 12 \))
• Adjacent side to \( A \): 5 (assuming \( AC = 5 \))

Step3: Calculate \( \tan(A) \)

Using the definition \( \tan(A)=\frac{\text{opposite}}{\text{adjacent}} \), we substitute the lengths of the opposite and adjacent sides. So \( \tan(A)=\frac{12}{5} \).

Answer:

\(\frac{12}{5}\) (corresponding to the option with \(\frac{12}{5}\))