Why do we write radian measures with a π rather than just give the decimal equivalent? (for example, why not use 1.88 rad instead of 3π/5 rad?)
Thx
The use of π in expressing angle measures in radians is closely tied to the nature of circles and trigonometry. Radian measure is a way to express angles based on the relationship between the length of an arc and the radius of a circle.
One radian is defined as the angle subtended by an arc of length equivalent to the radius of the circle. This means that if you were to take a circle with radius 1, one full revolution around the circle would have an arc length of 2π. Therefore, 2π radians is equivalent to a full revolution or 360 degrees.
Using radians makes certain mathematical properties and relationships more apparent. Some key reasons we use radians instead of decimal equivalents are:
1. Simplicity: Radians simplify calculations involving angles because they eliminate arbitrary conversions between degrees and more complex decimal values. Radians provide a natural way to link the geometry of a circle with trigonometric functions.
2. Consistency: Expressing angles in radians allows for consistency across various mathematical formulas and identities involving trigonometric functions. Many important formulas and relationships, such as the sine, cosine, and tangent functions, are derived and best understood when angles are measured in radians.
3. Connections to calculus: Radian measure aligns naturally with calculus and simplifies trigonometric calculations when dealing with derivatives and integrals. These connections are particularly important in physics, engineering, and other fields that heavily rely on calculus-based calculations.
While it is possible to express angles in decimal equivalents of radians, using π allows for a more concise and standardized notation that simplifies mathematical reasoning and consistency. It also facilitates connections to other mathematical concepts and applications, making radians a preferred choice in many situations.