Given that Cos A = \(\frac{12}{13}\) for 0 ≤ A ≤ 90º, find Tan A.

Answer Details

We are given that cosine of angle \(A\) is \(\frac{12}{13}\), and \(A\) is in the range \(0^\circ \leq A \leq 90^\circ\) (so all trigonometric values are positive).

To find \(\tan A\), let's recall the relationships among trigonometric functions for a right triangle:

• \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\)
• The Pythagorean identity: \(\sin^2 A + \cos^2 A = 1\)

Since \(\cos A = \frac{12}{13}\), if we imagine a right triangle:

• The length of the side adjacent to angle \(A\) is 12.
• The hypotenuse is 13.

Let's find the length of the opposite side using the Pythagorean theorem:

\[ \text{opposite}^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2 \] \[ \text{opposite}^2 + 12^2 = 13^2 \] \[ \text{opposite}^2 + 144 = 169 \] \[ \text{opposite}^2 = 169 - 144 = 25 \] \[ \text{opposite} = \sqrt{25} = 5 \]

Now we can find \(\tan A\):

\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12} \]

Therefore, the correct value of \(\tan A\) is \(\frac{5}{12}\).

This comes from using the relationships in a right triangle, plus the Pythagorean theorem to find all sides of the triangle when one trigonometric ratio is known. Whenever you know cosine (adjacent/hypotenuse), you can always find sine (opposite/hypotenuse) using the Pythagorean identity, and then use those to find tangent (\(\tan = \frac{\sin}{\cos}\)).