Trigonometric Functions of Any Angle

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Math Analysis - Unit Circle - How to create the Unit Circle from scratch

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1 of 2 videos by AJ Speller

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)=theta-theta^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)=0.523-0.024+3.26*10^(-4)-...=0.499approx0.5#

    You can find the Taylor series for the other trigonometric functions such as:
    enter image source here
    (Picture source: www.efunda.com)

  • The reference angle is the smallest angle you can make between the x-axis and the terminal side of an angle. You always look at the x-axis 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 x-axis to that terminal side.

    If # 90< x<180#, the terminal side is in the second quadrant, the reference angle is equal to #180-x#.

    If # 180< x<270#, the terminal side is in the third quadrant, the reference angle is equal to #x-180#.

    If #270< x<360#, the terminal side is in the fourth quadrant, the reference angle is equal to #360-x#.

  • 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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