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
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
