Why do we write radian measures with a π rather than just give the decimal equivalent?

So that the equations can be exact.

For example,
Tan (pi/4) = 1 is exact

We write radian measures using the symbol π because radians are a more natural and fundamental way to measure angles in mathematics. Radians are defined based on the concept of a circle, where the measure of an angle in radians is equal to the length of the arc on the unit circle subtended by that angle.

The use of π in radian measures is rooted in the relationship between the circumference and the diameter of a circle. The value of π, approximately equal to 3.14159, is an irrational number, meaning it cannot be expressed exactly as a fraction or a decimal. It is a fundamental constant in mathematics that appears in various mathematical formulas and relations.

Using π in radian measures allows us to express angles in a way that is independent of the specific size of the circle being considered. Radians provide a consistent and more precise way to describe angles, especially in mathematical expressions and calculations.

To convert an angle measure from radians to its decimal equivalent, you can simply multiply the radian measure by the decimal approximation of π (usually rounded to a desired precision) or by using a calculator with a built-in π button. Conversely, to convert from degrees to radians, you can divide the degree measure by 180 and multiply by π.