Section 7-3
The Sine and Cosine Functions
Sine
• The “sin” that appears in the equation is
an abbreviation of the sine function, one of
the two trigonometric functions that we will
discuss in this section.
Cosine
• The “cos” that appears in the equation is
an abbreviation of the cosine function, the
other trigonometric function that we will
discuss in this section.
Sine and Cosine
Suppose P(x, y) is a point
2
2
2
on the circle x  y  r
and Θ is an angle in
standard position with
terminal ray OP as
shown.
• We define the sine of Θ,
y
sin


denoted sin Θ by:
O
r
• We define the cosine of
x
cos


Θ, denoted cos Θ by:
r
Sine and Cosine
• Although the definitions of sin Θ and cos Θ
involve the radius r of a circle, the values
of sin Θ and cos Θ depend only on Θ.
• Do activity on p. 269.
The Unit Circle
x2  y2  1
• The circle
has radius 1 and is
therefore called the unit circle. This circle
is the easiest one with which to work
because, as the diagram shows, sin Θ and
cos Θ are simply the y- and x-coordinates
of the point where the terminal ray of Θ
intersects the circle.
Sine, Cosine, and the Unit Circle
y
sin   
r
x
cos   
r
y
y
1
x
x
1
The Unit Circle
• Remember: The unit
circle has a radius of 1.
• Label the axes in both
radians and degrees.
• Label the points with the
x and y coordinates.
• Use the diagram to find
the values for sin or
cosine.
Circular Functions
• When a circle is used to define the
trigonometric functions, they are
sometimes called circular functions.
Domain
• From the definitions and diagram on the
preceding slides, we can see that the
domain of both the sine and cosine
functions is the set of all real numbers,
since sin Θ and cos Θ are defined for any
angle Θ.
Range
• The range of both functions is the set of all
real numbers between -1 and 1 inclusive,
since sin Θ and cos Θ are the coordinates
of points on the unit circle.
Sine and Cosine
• The diagrams below indicate where the sine and cosine
functions have positive and negative values. For
example, if Θ is a second-quadrant angle, sin Θ is
positive and cos Θ is negative.
Sine and Cosine
• You can remember the signs by
remembering “All Students Take
Calculus.”
All trig
functions
positive
Sine is
positive
Students
Tangent is
positive
Take
All
Calculus
Cosine is
positive
Values of Sine and Cosine
• The sine and cosine functions repeat their
values every 360˚ or 2π radians. Formally
this means that for all Θ:
sin   360  sin 
cos  360  cos 
sin   2   sin 
cos  2   cos 
Periodic Functions
• We summarize these facts by saying that
the sine and cosine functions are periodic
and that they have a fundamental period
of 360˚ or 2π radians. It is the periodic
nature of these functions that makes them
useful in describing many repetitive
phenomena such as tides, sound waves,
and the orbital paths of satellites.
Example
• Find the value of each expression without
using a calculator or table.
a. Cos 90°
b. Sin
c. sin 180 °
d. Cos π
Example
1. State whether the sine and the cosine of
each angle is positive, negative, or zero.
2

3
3π
Example
Complete each statement using <, >, or =.
• sin 30˚ _____ sin (-30˚)
• cos 30˚_____ cos (-30˚)
• cos 300˚ _____ cos 330˚