1. The problem asks for the definitions of the tangent and cosine functions in trigonometry. 2. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The formula is: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. The cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. The formula is: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 4. These definitions are fundamental in trigonometry and are used to relate angles to side lengths in right triangles. 5. Remember that these ratios are only valid for angles in right triangles or can be extended to all angles using the unit circle definition.