Involutes and Gear Ratios:
In mechanical engineering, involute gears are commonly used to transmit power between two rotating shafts. The gear ratio is an important parameter that determines the relative speeds and torques of the input and output shafts. The gear ratio is defined as the ratio of the number of teeth on the output gear to the number of teeth on the input gear.

Pitch Circle and Base Circle:
In an involute gear, the pitch circle is an imaginary circle that passes through the point of tangency between two gears. It is defined as the circle from which the gear tooth profiles are derived. The base circle, on the other hand, is the circle on which the involute curve is generated. It is defined as the circle that is tangent to the tooth profile at the base of the gear tooth.

Pressure Angle:
The pressure angle is a parameter that determines the shape of the gear tooth. It is the angle between the line of action, which is the line along which the teeth of two gears contact each other, and the tangent to the pitch circle at the point of contact.

Ratio of Pitch Circle Radius to Base Circle Radius:
The ratio of the pitch circle radius to the base circle radius in an involute gear is given by the cosine of the pressure angle. This can be mathematically expressed as:

Pitch Circle Radius / Base Circle Radius = cos(f)

where f is the pressure angle.

Explanation of the Correct Answer:
The correct answer is option 'B' - cos(f). This is because the ratio of the pitch circle radius to the base circle radius in an involute gear is given by the cosine of the pressure angle.

The cosine function is a trigonometric function that relates the ratio of the length of the adjacent side of a right triangle to the length of the hypotenuse. In the context of involute gears, the pressure angle is the angle between the line of action and the tangent to the pitch circle at the point of contact. The line of action is the adjacent side of the right triangle, and the tangent to the pitch circle is the hypotenuse.

By taking the cosine of the pressure angle, we obtain the ratio of the pitch circle radius to the base circle radius. This ratio is an important parameter in gear design as it affects the gear tooth profile and the gear ratio.

In conclusion, the ratio of the pitch circle radius to the base circle radius in an involute gear is given by the cosine of the pressure angle.