Ch. 5
Trigonometric
Functions of
Real Numbers
Melanie Kulesz
Katie Lariviere
Austin Witt
THE UNIT CIRCLE
x2 + y2 = 1
The circle of radius 1 centered at the origin in the xy-plane.
Proving points on the unit
circle


Use equation: x2 + y2 = 1
See Example
LOCATING A POINT ON THE CIRCLE
See example
 Use equation: x2 + y2 = 1

Terminal Points
Terminal Point – the point P(x,y) obtained
and determined by the real number t
 Suppose t is a real number. Mark off a
distance t along the unit circle, starting at
the point (1,0) and moving in a
counterclockwise direction if t is positive or
in a clockwise direction if t is negative
 See example
t = -π

Reference Numbers
 Reference Number - the shortest distance
along the unit circle between the terminal point
determined by t and the x-axis
 Sine Curve
 Cosine Curve
 Tangent Curve
 Stretch
 Shift
 Amplitude
 Period
To find the terminal point P determined by any
value of t, use the following steps…
1. Find the reference number t
2. Find the terminal point Q(a, b) determined
by t
3. The terminal point determined by t is P(±a,
±b ), where the signs are chosen according
to the quadrant in which this terminal point
lies
See Example
Trigonometric Functions
sin t = y
cos t = x
tan t = y/x (x≠0)
csc t = 1/y (y≠0)
sec t = 1/x (x≠0)
cot t = x/y (y≠0)
See example
Even-Odd Properties






Sin(-t) = -sin t
Cos(-t) = cos t
Tan(-t) = -tan t
Csc(-t) = -csc t
Sec(-t) = sec t
Cot(-t) = -cot t
Odd
Even
Quadrant Positive Functions
I
II
III
IV
all
sin, csc
tan, cot
cos, sec
Negative
functions
none
cos, sec, tan, cot
sin, csc, cos, sec
sin, csc, tan, cot
SIGNS OF THE
TRIGONOMETRIC FUNCTIONS
Fundamental Identities
● Reciprocal Identities:
csc t = 1/sin t
sec t = 1/cos t
cot t = 1/tan t
tan t = sin t/cos t
cot t = cos t/sin t
● Pythagorean Identities:
sin^2t + cos^t = 1
tan^2t + 1 = sec^2t
1 + cot^2t = csc^2t
Trigonometric Graphs
• Periodic Properties:
The functions tan and cot have period
π
tan(x + π) = tan x
cot(x + π) = cot x
The functions csc and sec have period
2π
csc(x + 2π) = csc x
sec(x + 2π)= sec x
•
•
The functions of Sine and Cosine both have a
period of 2π
This means they repeat themselves after one full
rotation around the unit circle
THE FUNCTION OF SINE
•
•
•
•
•
The sine function starts from the origin
It then follows the pattern of Peak, Root, Valley
The roots are at every 1 Pi when the period is 2 Pi
The peaks are equal to the amplitude which is equal to the coefficient
of the function.
Valleys are also derived from the amplitude
F(x)=sinx
The Function of Cosine
•
•
•
•
•
The Cosine Function starts at a peak which is equal to
amplitude or coefficient of the function.
It then follows the pattern root, valley, peak. The roots
occurring at every 1/2Pi.
The valleys and peaks equal to the amplitude.


Horizontal stretching occurs when you a have a change of
the period of the function.

Ex 1. sin2x would repeat itself twice in the one rotation of the unit
circle.

Ex2. sin1/2x would repeat itself once in 2 rotations of the unit
circle.
Vertical stretching occurs from a change in amplitude or the
coefficient of
 function.


Ex 1. 2sinx would have a peak and valley at 2 and -2 respectively.
Horizontal
shifts of the sine and
cosine functions are shown as
sin(x+a)
where is some value in radians.
Vertical
shifts look like sinx+a which
would move it up or down
depending on (a).
The Tangent Function
The tangent function has a period of Pi but starts out at
negative ½ Pi
 and goes to positive ½ Pi.
• Its shape liked an “s” and intersects the origin in the
middle
• It also has asymptotes' at the beginning and end of each
period
•











Cotangent = 1/tan : the reciprocal of tangent starts at the origin with an
asymptotes at the origin and has a period of 1 Pi where it ends with
another
asymptote. It too looks like an “s” but it has a negative slope as it moves
from
Positive infinity to negative infinity in its “Y” values.
Cosecant =1/sin : the reciprocal of sine has asymptotes at every ½ Pi .
If you take the peaks of the cosine function that is the vertex of the
Parabola formed by the reciprocal
Secant= 1/cos: the reciprocal of the cosine function is related to the
Cosecant function in that its parent function’s peaks are the vertices of the
Parabolas formed. However secant has asymptotes at 0 and 1 Pi instead
Of every ½ pi.