- #31
sunny79
- 77
- 8
@hutchphd! Still here...just terribly busy.
Why does the radius of a unit circle need to be 1?
- Context: High School
- Thread starter sunny79
- Start date
-
- Tags
- Circle Radius Unit Unit circle
Click For Summary
SUMMARY
The radius of the unit circle is defined as 1 to simplify the teaching of trigonometric functions and radians. This definition allows for the circumference to be expressed as 2π radians, facilitating easier calculations in trigonometry. The unit circle's radius being 1 also means that the sine and cosine of angles correspond directly to the coordinates of points on the circle, enhancing understanding of these functions. The choice of 1 as the radius is a matter of convention, providing a standard reference point for mathematical calculations.
PREREQUISITES- Understanding of basic trigonometric functions (sine, cosine)
- Familiarity with radians and their relationship to angles
- Knowledge of the properties of circles, including circumference and area
- Basic mathematical conventions and definitions in geometry
- Explore the concept of radians in depth, focusing on their application in trigonometry
- Study the historical development of the unit circle and its significance in mathematics
- Learn about the implications of using different radii in trigonometric calculations
- Investigate the debate surrounding the use of π versus τ in mathematical contexts
Mathematics educators, students learning trigonometry, and anyone interested in the foundational concepts of geometry and trigonometric functions.
- #32
sysprog
- 2,617
- 1,793
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:gmax137 said:
- #33
Mark44
Mentor
- 38,075
- 10,598
This type of caliper would work like the Vernier caliper shown in post #29. If the jaws of this caliper are set too close, the circle wouldn't fit between the jaws. Opening the jaws to a width so that they just barely accept the circle would give the diameter.sysprog said:
I'm assuming we have an object with circular cross section here, although you could also do the measurement reasonably well for a circle drawn on paper. Put one jaw at any point on the circumference, and adjust the caliper opening so that the other jaw intersects a single point as the caliper is rotated through an arc.
- #34
gmax137
Science Advisor
Education Advisor
- 3,152
- 3,715
I'm sorry, my post #26 was a distraction, intending to address the distraction that started around post #9, debating radius vs. diameter, and pi vs. 2*pi.
- #35
sysprog
- 2,617
- 1,793
Isn't that still a series of trials? How do we find that we haven't exceeded the diameter? Don't we have to do repeated trials to find out exactly where "just barely" is?Mark44 said:
- #36
sysprog
- 2,617
- 1,793
This method makes good sense to me.Mark44 said:
- #37
mathwonk
Science Advisor
Homework Helper
2024 Award
- 11,959
- 2,233
of course the assumption that caliper measure equals ruler measure is equivalent to the Side angle side axiom of euclidean geometry.
Pardon me, I am entranced by a thread that goes on this long, in answer to OP's question of "why is 1 = 1?"
Pardon me, I am entranced by a thread that goes on this long, in answer to OP's question of "why is 1 = 1?"
Similar threads
- · Replies 19 ·
- · Replies 5 ·
- · Replies 59 ·
- · Replies 7 ·
Undergrad
Proof of average height of half circle
- · Replies 19 ·
- · Replies 2 ·
- · Replies 6 ·
- · Replies 6 ·
- · Replies 8 ·