Trigonometric Functions of Any Angle
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Key Questions

Well, I guess you could use a special representation of the function through a sum of terms, also known as Taylor Series.
It is, basically, what happens in your pocket calculator when you evaluate, for example,#sin(30°)# .
Your calculator does this:
#sin(theta)=thetatheta^3/(3!)+theta^5/(5!)...#
where#theta# must be in RADIANS.
In theory you should add infinite terms but, depending upon the accuracy required, you can normally stop at three terms.
In our case we have:#theta=pi/6=3.14/6=0.523# and:
#sin(pi/6)=sin(0.523)=0.5230.024+3.26*10^(4)...=0.499approx0.5# You can find the Taylor series for the other trigonometric functions such as:
(Picture source: www.efunda.com) 
The reference angle is the smallest angle you can make between the xaxis and the terminal side of an angle. You always look at the xaxis as your frame of reference.
For example, for an angle measuring
#x# degrees,if
#x<90# , the terminal side is in the first quadrant, the reference angle is the acute angle formed counterclockwise from the xaxis to that terminal side.If
# 90< x<180# , the terminal side is in the second quadrant, the reference angle is equal to#180x# .If
# 180< x<270# , the terminal side is in the third quadrant, the reference angle is equal to#x180# .If
#270< x<360# , the terminal side is in the fourth quadrant, the reference angle is equal to#360x# . 
The good thing about a unit circle is the fact that any point on it already has the coordinates
#(cosx, sinx)# . This comes from the fact that#x=rcos# and#y=rsinx# , and that#r=1# for the unit circle. From this we can easily get the values for#tanx# ,#cotx# ,#secx# , and#cscx# .
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