Section 7-4
Evaluating and Graphing Sine and
Cosine
Copy and Complete the chart
Sign of:
Quadrant
I
II
III
IV
x
y
Cos Θ
Sin Θ
Reference Angles
• Let α (Greek Alpha) be an acute angle in
standard position. Suppose that α = 20°.
Notice that the terminal ray of α = 20° and
the terminal ray of 180° - α = 160° are
symmetric in the y-axis. If the sine and
cosine of α = 20° are known, then the sine
and cosine of 160° can be deduced.
Reference Angles
x2  y2  1
Sin 160° = y = sin 20°
Cos 160° = x = - cos 20°
Reference Angle
• The angle α = 20° is called the reference
angle for the 160° angle. It is also the
reference angle for the 200° and 340°
angles shown below.
Reference Angles
• In general, the acute angle α is the
reference angle for the angles 180° - α,
180° + α, and 360° - α as well as all
coterminal angles. In other words, the
reference angle for any angle Θ is the
acute positive angle formed by the
terminal ray of Θ and the x-axis.
Reference Angles
• Is the angle between
0° and 360° (or 0 and
2π)?
• What quadrant is the
angle in?
• Is sin or cos positive
or negative in that
quadrant?
• Apply the reference
angle formula.
180°- Θ
π-Θ
Θ - 180°
Θ-π
Θ
360°- Θ
2π - Θ
Determine the Sign
** The sine or cosine of the reference angle
for a given angle gives the absolute value
of the sine or cosine of the given angle.
The quadrant in which the given angle lies
determines the sign of the sine or cosine
of the given angle.
Using Calculators or Tables
• The easiest way to find the sine or cosine
of most angles is to use a calculator that
has the sine and cosine functions. Always
be sure to check whether the calculator is
in degree or radian mode.
• If you do not have access to a calculator,
there are tables at the back of the book.
Instructions on how to use the tables are
on p. 800
Finding Sines and Cosines of
Special Angles
• Because angles that are multiples of 30°
and 45° occur often in mathematics, it can
be useful to know their sine and cosine
values without resorting to a calculator or
table. To do this, you need to know the
following facts:
Finding Sines and Cosines of
Special Angles
In a 30°-60°-90°
triangle, the sides are
in the ratio 1 : 3 : 2 (note
that in this ratio, 1
corresponds to the
side opposite the 30°
angle, 3 to the side
opposite the 60°
angle, and 2 to the
side opposite the 90°
angle.)
Finding Sines and Cosines of
Special Angles
In a 45°-45°-90°
triangle the sides
are in the ratio1 : 1 : 2
, or 2 : 2 : 2 .
• These facts are used in the diagrams
below to obtain the values of sin Θ and
cos Θ for Θ = 30°, Θ = 45°, and Θ = 60°.
Special Angles
• Although the table only gives the sine and
cosine values of special angles from 0° to
90°, reference angles can be used to find
other multiples of 30° and 45°.
1
• For example: sin 210   sin 30  
2
2
cos 315  cos 45 
2
Special Angles
Special Angles
• As the table suggests, sin θ and cos θ are
both one-to-one functions for the first
quadrant θ
Graphs of Sine and Cosine
• To graph the sine function, imagine a particle on the unit
circle that starts at (1, 0) and rotates counter clockwise
around the origin. Every position (x, y) of the particle
corresponds to an angle θ, where sin θ = y by definition.
As the particle rotates through the four quadrants, we get
the four pieces of the sine graph shown below.
• I From 0° to 90°, the y-coordinate increases from 0 to 1.
• II From 90° to 180°, the y-coordinate decreases from 1 to
0.
• III From 180° to 270°, the y-coordinate decreases from 0
to -1.
• IV From 270° to 360°, the y-coordinate increases from -1
to 0.
• Copy graph on p. 278
• Use the unit
circle to
complete the
table of
values.
• Then use the
table of values
to graph sine
and cosine
The Sine Graph
The Sine Graph
The Sine Graph
• Since the sine
function is periodic
with a fundamental
period of 360°, the
graph above can be
extended left and
right as shown below.
The Cosine Graph
The Cosine Graph
The Cosine Graph
• To graph the cosine
function, we analyze
the x-coordinate of
the rotating particle in
a similar manner.
The cosine graph is
shown below.
Example
• Express each of the following in terms of a
reference angle.
a.Sin 120°
b.Cos (-120°)
c.Sin 690°
Example
• Use a calculator or table to find the value
of each expression to four decimal places.
a.Sin 43°
b.Cos 55°
c.Cos 230.46°
d.Sin 344.1°
Example
• Give the exact value of each expression in
simplest radical form.
a.Sin 30°
b. Sin
c. Cos (-120°)
d. cos
Example
1. Find the exact value of each expression:
7
sin
6
7
cos
6
log 4 sin 150
Example
x
• Sketch the graph of y = - sin x and y = 3
on the same set of axes. How many
x
solutions does the equation –sin x =
3
have?