2a4 Math Website
  • Home
    • Basics
    • Methods of Proof>
      • Induction
  • Algebra
    • Binomial Theorem
    • Arithmetic Progression
    • Factorization/Expansion
    • Functions>
      • Absolute Value
      • Logarithms
    • Algebraic Manipulation
    • Inequalities
    • Linear/ Quadratic>
      • Linear Diophantine Equations
      • Quadratic Surds
      • Roots, Coefficients, and Discriminants of Quadratic Expressions
    • Polynomial>
      • Basic Formulas on Polynomials
      • Division of Polynomials
    • Matrices
  • Number Theory
    • Perfect Squares
  • Geometry
    • Geometrical Properties of Circles
    • Midpoint Theorem
    • Triangles>
      • Congruence of Triangles
      • Similarity of Triangles
      • Properties of Triangles and Angles
      • Area & Perimeter of Triangles
      • Centers of a Triangle
      • Basic Trigonometry
    • Radians
    • Intro to Solids
  • SMO
    • SMO 2012 Round 2 Solutions
    • Introduction
  • Weekly Questions
    • Week 1-10>
      • Week 1>
        • Week 1 Solutions
      • Week 2>
        • Week 2 Solutions
      • Week 3>
        • Week 3 Solutions
      • Week 4>
        • Week 4 Solutions
      • Week 5>
        • Week 5 Solutions
      • Week 6>
        • Week 6 Solutions
      • Week 7>
        • Week 7 Solutions
      • Week 8>
        • Week 8 Solutions
      • Week 9>
        • Week 9 Solutions
      • Week 10>
        • Week 10 Solution
    • Week 11>
      • Week 11 Solution

Radians

Radians are one way of calculating angles.
There are 4 common ways:
 - Radians
- Degrees
- Gradians (basically, 360 degrees = 400 gradians)
- turns - 360 degrees = a turn


From this, we can see that the most complicated type, is in fact, radians.
A radian is the ratio between the length of an arc and it's radius.


i.e. ----------> length of subtended (by radius) arc / radius ---------------> s / r in the diagram below)----------------------> gives us radians


Hence, you get 1 radian or "1 rad" when length of subtended arc = radius
Picture
From the picture, 
We see that s is the subtended arc and AO and BO subtend it. That's what subtend means... to like erm... "mark the boundary of"
When s = r, s/r = r/r = 1 radian


Now, many people say that 2pi = 360 degrees in trigonometry. But why is this so? 
This is explained by the derivation or radians.
Take a look at the diagram again. Notice that the angle controls the size of s, the subtended arc.


When tita, the angle is bigger, s is also bigger. When tita is smaller, s will also be smaller. The angle and subtended arc are thus proportional.


Think about it this way,
when tita is 90 degrees, AOB will be a quarter-circle --- making s a quarter of the total circumference.


Let's jump a bit here. 
By definition, 
Since 2(pi)r = circumference of a circle, where according to the diagram, tita = 360. Here, we see that y = 2(pi)r
2(pi)r / r radians = 2(pi) radians

Therefore, 2(pi)radians = 360 degrees.

Dividing 2(pi) on both sides gives

1 radian = 360/2(pi) = 180/pi

Hence,180 = pi, 360 = 2(pi)


Powered by Create your own unique website with customizable templates.