Finding the values (or ratios) of
sine and
cosine for the
unit circle is very easy given that you can
memorize a simple table of values. The table for cosine goes like this:
cos0 = sqrt4/2 = 1
cos30 = sqrt3/2
cos45 = sqrt2/2
cos60 = sqrt1/2 = 1/2
cos90 = sqrt0/2 = 0
If you notice, is starts with sqrt4/2 and with each of these
angles the value of the number having the
square root decreases by one in each step until it reaches 0.
Likewise, here is the table for sine. Sine's table is the
inverse of cosine's, so it should be easy to
remember.
sin0 = sqrt0/2 = 0
sin30 = sqrt1/2 = 1/2
sin45 = sqrt2/2
sin60 = sqrt3/2
sin90 = sqrt4/2 = 1
Because we know that
tangent is sine/cosine, we can find the value of tangent by putting the
exact value of sine or that of cosine. The 2s of the
denominator will always cancel out so you can just put the
numerator of sine over cosine's. In the same way, we know that
secant and
cosecant are the
inverse of cosine and sine so we can find these values also. Simply take the
reciprocal of the sine, cosine, or even tangent (for
cotangent) to find their inversed values. If the denominator need rationalizing, do so by multiplying by a form of 1.