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                                      Michael Lee's Mathematics and Physics

Trigonometry

 

The following animation describes what radians are; they are just a measure of angle much like degrees.  

All you need to know and remember is a full circle consists of exactly...

 

  

 

 

Parts of a Right Triangle 

Locating the hypotenuse is easy; not only is it the longest side but it is always "across from" the right angle of the triangle.  The opposite side is always "across from" angle theta and the adjacent side is "next to" angle theta.

 

 

If we change theta to this corner instead, then the opposite side and the adjacent side transpose each other.

 

 

 

The Unit Circle and All Other Circles

Examine the following diagram, angle theta is shown in a sample position, the inner circle (it's called the unit circle) has a radius of one whereas the outer circle has a said radius of r.  Because of the similar relationships between the two triangles, ABC and ADE, (i.e. they are similar right triangles) the following can be said,

 

 

 

These relationships still hold if the circle in question has a radius smaller than one. 

 

Line segments x and y, from the diagram above, can be expressed as functions of angle theta. Below is an animation of what happens to x and y as angle theta changes, as it goes from 0 to 1 pi, 2 pi, 3 pi and so on.  The values of the cosine and sine functions vary from negative one to positive one, vertically, for the unit circle and the horizontal axis is in radians where each "block" is equal to half of pi.