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Miike012
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Homework Statement
What made them pick one rad equal to 57 deg.
Why didn't the pick 45 deg. or 90 deg. or any other random degree to equal one rad.?
Why 57 Degrees Was Chosen for One Radian
- Thread starter Miike012
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SUMMARY
The choice of defining one radian as approximately 57 degrees stems from the relationship that π radians equals 180 degrees, making the exact conversion 180/π degrees. This definition simplifies various mathematical equations involving angles, such as arc length and angular velocity. By using radians, mathematicians avoid awkward constants in calculations, enhancing clarity and efficiency in mathematical expressions.
PREREQUISITES- Understanding of basic trigonometric functions
- Familiarity with the concept of radians and degrees
- Knowledge of angular motion equations
- Basic calculus principles, particularly derivatives
- Research the derivation of the relationship between radians and degrees
- Explore the applications of radians in physics, particularly in rotational dynamics
- Learn about the significance of π in mathematics and its applications
- Investigate the use of radians in calculus, especially in integration and differentiation of trigonometric functions
Students studying mathematics, physics enthusiasts, and anyone interested in understanding the significance of angular measurements in various scientific applications.
What made them pick one rad equal to 57 deg.
Why didn't the pick 45 deg. or 90 deg. or any other random degree to equal one rad.?
- #2
Gib Z
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Google "radians" and just read how they are defined. Whats important is that \pi radians is equal to 180 degrees, not that 1 radian is approximately 57 degrees.
- #3
tiny-tim
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Hi Miike012! 
It isn't exactly 57°, it's exactly (180/π)°.
It's to avoid having an awkward constant in various equations …
with angle θ measured in radians, we have the simple equations:
Also, with ω measured in radians per second:
This is similar to the reason mathematicians prefer natural logarithms (base e) to logarithms base 10.
(Similarly, we define the steradian to measure "solid angle", giving:
It isn't exactly 57°, it's exactly (180/π)°.
It's to avoid having an awkward constant in various equations …
with angle θ measured in radians, we have the simple equations:
arc-length = rθ (in particular, circumference = r(2π))
speed = r dθ/dt
acceleration = r d2θ/dt2
speed = r dθ/dt
acceleration = r d2θ/dt2
Also, with ω measured in radians per second:
d/dt (sinωt) = ωcosωt.
This is similar to the reason mathematicians prefer natural logarithms (base e) to logarithms base 10.
(Similarly, we define the steradian to measure "solid angle", giving:
surface area = r2 times solid angle)
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