## Friday, December 22, 2006

### 4.2 Trigonometric Functions: The Unit Circle

4.2 Trigonometric Functions: The Unit Circle

Unit Circle (x2 + y2)= 1

As the real number line is wrapped around the unit circle, each real number t corresponds to a point (x,y) on the circle. For example, the real number 0 corresponds to the point (1,0). Because the unit circle has a circumference of 2π, the real number 2π also corresponds to the point (1,0).
Examples:

1.) π/2 corresponds to (x,y) = (0, 1)

2.) π corresponds to (x,y) = (-1, 0)

3.) 3π/2 corresponds to (x,y) = (0, -1)

In general, each real number t corresponds to a central angle θ (in standard positition) whose radian measure is t.

Trigonometric Functions:
Definitions of Trigonometric Functions:
Let t be a real number and let (x,y) be the point on the unit circle corresponding to t.

sine t is written as sin t = y

cosecant t is written as csc t = 1/y

cosine t is written as cos t = x

secant t is written as sec t = 1/x

tangent t is written as tan t = y/x

cotangent t is written as cot t = x/y

 Degrees 0° 30° 45° 60° 90° Radians 0 radians π/6 π/4 π/3 π/2 sin θ cos θ tan θ csc θ sec θ cot θ

Withing the unit circle, you can see that :

A). the value of x is betweeen -1 and 1

B). the value of y is between -1 and 1

Definition of a Periodic Function
A function f is periodic if there exists a positive real number c such that
f (t + c) = f (t)
for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

Example:
26π/12 = 2π + 2π/12 , you have sin (26π/12 ) = sin (2π + 2π/12 ) = sin 2π/12 = ½

Even and Odd Trigonometric Functions
The cosine and secant functions are even.

cos ( -t) = cos ( t )

sec ( -t ) = sec ( t )

The sine, cosecant, tangent and cotangent are odd.

sin( -t ) = -sin t

csc ( -t ) = - csc t

tan ( -t ) = -tan t

cot ( -t ) = -cot t

Examples:

sin ( -30°) = -½ -sin (30°) = -1/2

cos ( -60°) = ½

cos (60°) = ½

tan ( -45°) = -1

tan (45°) = -1