What is a periodic function?
Periodic functions are those that repeat the same values at regular intervals. For trigono-
metric functions, the period(or the interval of repeated values) is 360 degrees or 2πfor
the sin, cosine, secant, and cosecant; and 180 degrees or πradians for the cotangent and
tangent. (For example, this is often said as “the tangent function has period π.”)What are the hyperbolic functions (or identities)?
Similar to the other types of functions and identities above, hyperbolic functions (also
called hyperbolic identities) are used to make it easier to find solutions to equations.
In fact, there are corresponding trigonometric and hyperbolic identities. For example,
the most commonly used trig identity, cos^2
sin^2
1, has a corresponding hyper-
bolic identity:
cosh^2 xsinh^2 x 1
In this case, the minus sign is used instead of a plus and the “cos” and “sin”
204 changed to “cosh” and “sinh,” respectively (also changed is the “^ ” symbol to x; this is
What is one of the most fundamental identities?
I
f one examines some of the functions above carefully, the sin and cosine func-
tions are actually the coordinates of a point on the unit circle. This implies
that the most important fundamental formula in trigonometry is as follows—
one that some people call the “magic identity” but is more commonly known as
one of the Pythagorean identities:cos^2
sin^2
1, in which^ is any real number.
This identity can be used in the following step-by-step example:
Show that:
sec^2
tan^2
^1
Because sec
1/cos (from the reciprocal identities), and tan
sin /cos
(from the ratio or quotient identities) then:
tan^2 ^1 (sin^ /cos^ )^2 ^1
(sin^2 /cos^2 ) 1
(sin^2
cos^2 )/cos^2 (where cos^2 divided by itself equals 1).Then (using the Pythagorean identity):
1/cos^2
sec^2 (from the reciprocal identity).