Applications of Radian Measure

Key Questions

  • What are some applications of using radian measure?

    In physics you use radians to describe circular motion, in particular you use them to determine angular velocity, #omega#.
    You may be familiar with the concept of linear velocity given by the ratio of displacement over time, as:
    #v=(x_f-x_i)/t#
    where #x_f# is the final position and #x_i# is the initial position (along a line).
    Now, if you have a circular motion you use the final and initial ANGLES described during the motion to calculate velocity, as:
    #omega=(theta_f-theta_i)/t#
    Where #theta# is the angle in radians.
    #omega# is angular velocity measured in rad/sec.
    enter image source here
    (Picture source: http://francesa.phy.cmich.edu/people/andy/physics110/book/chapters/chapter6.htm)

    Have a look to other rotational quantities you'll find a lot of ...radians!

    Gió · 1 · Jan 11 2015
  • How do you calculate the length of an arc and the area of a sector?

    For any #theta#, the length of the arc is given by the formula (if you work in radians, which you should:
    mathwarehouse.com
    The area of the sector is given by the formula #(theta r^2)/2#

    Why is this?
    If you remember, the formula for the perimeter of a circle is #2pir#.
    In radians, a full circle is #2pi#. So if the angle #theta = 2pi#, than the length of the arc (perimeter) = #2pir#. If we now replace #2pi# by #theta#, we get the formula #S = rtheta#

    If you remember, the formula for the area of a circle is #pir^2#.
    If the angle #theta = 2pi#, than the length of the sector is equal to the area of a circle = #pir^2#. We've said that #theta = 2pi#, so that means that #pi = theta/2#.
    If we now replace #pi# by #theta/2#, we get the formula for the area of a sector: #theta/2r^2#

    Nico Ekkart · 5 · Dec 19 2014
  • How do you approximate the length of a chord given the central angle and radius?

    Let's call the cord #AB# and the centre of the circle #C#

    Then if you divide the cord in half at #M# you get two equal, but mirrored triangles #Delta CMA# and #Delta CMB#. These are both rectangular at #M#. (You should draw this yourself right now !).

    #angle ACM# is half the central angle that was given
    (and #angleBCM#is the other half)

    Then #sin angle ACM=(AM)/(AC) ->AM=AC*sin angle ACM#

    Since you know the radius #(AC)# and the central angle (remember #angleACM=#half of that), you just plug in these values to get an accurate result for half the chord (so don't forget to double it for your final answer)

    MeneerNask · 1 · Jan 13 2015
  • What are some example problems involving angular speed?

    Answer:

    See examples in explanation

    Explanation:

    Earth's day/night spin about it axis is with

    angular speed = #2pi# radian / 24-hour day.

    Earths revolution about Sun is owith

    angular speed = #2pi) radian / 365.26-day year.

    Rotors making electro-mechanical rotations have high angular

    speeds of

    #kKpi# radian / minute, k > 1,

    making thousands of rpm ( revolutions / minute ).
    ,

    A. S. Adikesavan · 1 · Aug 10 2018

Questions

Graphing Trigonometric Functions