Finding Triangle Sides

We are given a right-angled triangle ABC, with $\angle B = 90^\circ$. The sides provided are AB = 12 cm (adjacent to A, opposite to C) and AC = 13 cm (hypotenuse).

We need to find the length of the side BC (opposite to A, adjacent to C) using the Pythagorean theorem: $AC^2 = AB^2 + BC^2$.

Substituting the values:

$13^2 = 12^2 + BC^2$

$169 = 144 + BC^2$

$BC^2 = 169 - 144$

$BC^2 = 25$

$BC = \sqrt{25} = 5 \text{ cm}$

Calculating Trigonometric Ratios

Now we calculate the required trigonometric ratios for angle A and angle C.

• Tangent of A ($\tan A$): This is the ratio of the side opposite to angle A to the side adjacent to angle A. $ \tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} = \frac{5}{12} $
• Cotangent of C ($\cot C$): This is the ratio of the side adjacent to angle C to the side opposite to angle C. $ \cot C = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{BC}{AB} = \frac{5}{12} $

Evaluating tan A - cot C

Finally, we compute the value of $\tan A - \cot C$.

$ \tan A - \cot C = \frac{5}{12} - \frac{5}{12} $

$ \tan A - \cot C = 0 $

Thus, the value of $\tan A - \cot C$ is 0.