What are Radians? | Radian (Unit of Plane Angle)

 

What are Radians?

                                                                           

          

A right angle measures 90º A shallow angle (half a turn) measures 180º And a complete 360º lap Do you know why exactly 360? It is an arbitrary number, which has to do with the base 60 number system Is there another way to measure angles that makes a little more sense? Yes ! This brings us to the concept of radians, and believe me, many people don't understand what it means. Most know how to convert degrees to radians and vice versa, but it ends there To really understand what radian means, let's draw a circle Be your center and your radius Now I'm going to take this radius measurement, and put it like this Now I'm going to double over the circumference So the length of this red arc is still r, it just doubled and nothing more The measurement arc r starts here and ends here Now, starting from the center, we draw another ray that connects at the other end of the red arc This measure is also worth r since it is a radius, let's analyze the figure We have an arc whose measure is the radius of the circle And two rays connecting the center and the ends of that arc Remember that the length of the arc is equal to the radius of the circle This angle then formed is worth 1 rad That's the definition of radian In fact, there are rays everywhere around that angle And that's probably why it was called radian According to many scientists and mathematicians this is a better way to measure angles because we are measuring the angle in relation to the radius of the circle But how is radian related to degree? How much would this angle be worth in degree? If you understood the definition of radian well, then the conversion between degree and radian will be easy What is the length of a circle? 2pi.r where r is radius of the circumference We are talking about the circumference, which is the outline of the circle Now one more question ... How many of these arches (red) will you need to cover the circumference exactly? I will repeat the question How much of these arcs measuring r will you need to cover the outline of the circle Pause the video and see if you can find out The arc length is r, and the circumference is 2pi.r, yes 2pi.r This means that we will need 2pi arcs of this to cover the circumference 2pi arcs? What does that mean ? Since pi is approximately 3.14, we need about 6.28 arcs to cover the circumference How would it be ? Here is 1 arc, 2, 3, 4, 5, 6, and a little piece measuring + - 0.28 of the arc, or 28% of the radius We then need 2pi arcs to cover the entire circumference And how much each corresponds from the center We saw that an arc measuring r corresponds to 1 rad in the center This implies that to complete a complete lap, we will have 2pi rad And even more If we rotate a radius completely, then we will cover 2pi radians in the center But wait, look We also know that a complete lap is worth 360º Yes, a full lap is worth 360º, as both are equivalent to a full lap, they will be the same This is the relationship between radian and degree What we wanted to get to in this video If we understand this, we can derive the rest This is a complete lap, so when is it worth 180º in rad? If we divide this equation by two, we have: E 180º equals half a turn, shallow angle How about a right angle? If we divide by 2 again we have: So our right angle is equal to pi over 2 rad These three relations are the most important for conversion between degree and radian And we don’t need to decorate the 3, knowing one or the definition is enough to deduce the rest And now we come to the last and most important part of the video How much is 1 rad in degree worth? Dividing the first relation by 2pi we have: Approaching pi by 3.14, we arrive that 1 rad is close to 57.29º This is the measure in degree of 1 rad If we have an angle like this, which is worth 1 rad, it will be close to 57.2958º. 

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