Let t be any real number and let P(x,y) be the terminal point on the unit circle defined by t. Then we define six functions of t:
sin(t) = y; “sine of t.”
cost(t) = x; “cosine of t.”
tan(t) = (y/x); “tangent of t.”
cot(t) = (x/y); “cotangent of t.”
sec(t) = (1/x); “secant of t.”
csc(t) = (1/y); “cosecant of t.”
These are called the trigonometric functions.
For any number of t, the terminal point determined by t is (cos(t), sin(t)).
Notice from this: That cos(t) & sin(t) are always between -1 and 1.
“Think of the other four trig functions in terms of sine and cosine.”
There are lots of inter-relationships between the trigonometric functions. We represent these by identities. An identity is an equation that’s true for every value of the variable for which it’s defined.
Remember from days of Algebra past: “An odd function has f(-t) = -f(t) for all ‘t’.”
and
“An even function has f(-t) = f(t) for all ‘t’.”
Now, there are Pythagorean Identities:
Remember that for any number ‘t’, (cos(t), sin(t)) is a point on the unit circle, and remember that the equation of the unit circle is x2 + y2 = 1.
This gives us the identity: - cos2(t) + sin2(t) = 1.
Along with the new identity, we also have two related identities: - cot2(t) + 1 = csc2(t)
Reciprocal Identities:
sec(t) = (1/cos(t))
csc(t) = (1/sin(t))
cot(t) = (1/tan(t))
these can also be written as:
cos(t) = (1/sec(t))
sin(t) = (1/csc(t))
tan(t) = (1/cot(t))
Odd-Even Identities:
Four of our trigonometric functions are odd functions:
sin(-t) = -sin(t)
csc(-t) = -csc(t)
tan(-t) = -tan(t)
cot(-t) = -cot(t)
Two of them are even functions:
cos(-t) = cos(t)
sec(-t) = sec(t)