**4.2 Trigonometric Functions: The Unit Circle**

Unit Circle (x^{2} + y^{2})= 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

same answer

cos ( -60°) = ½

cos (60°) = ½

same answer

tan ( -45°) = -1

tan (45°) = -1

same answer

sin ( -330°) = ½

sin (330°) = -( -½) = ½

same answer

Evaluating Trigonometric Functions with a Calculator

Make sure you set your calculator to the proper mode of measurement (degrees or radians). Also remember to enclose all fractional angle measures in parentheses.

Examples:

cos (5π/3) = 1/2