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4.3
Trigonometry: The Circular Functions
Initial Side – beginning position of the ray of an angle
Vertex – common endpoint of two rays that form an
angle
Terminal Side –final position of second ray in an angle
Positive angles go counterclockwise rotation
Negative angles are generated by clockwise rotations.
terminal side

initial side
Standard position – place angle with the vertex of the
angle at the origin and its initial side lying along the
positive x-axis.
4
2
-10
-5
-2
2
-4

-10
-5
5
positiv e angle
(counte rclockwise )
-2
-4
5

4
10
negativ e angle
(clockwise)
10
Notes 4.3 pre-calc page 2
Coterminal angles – angles that have same initial side
and the same terminal side yet have different measures.
4
2

-10
-5
5

10
4
positiv e and ne gative
cote rminal angles
-2
2

-4
-10

-5
5
two positiv e
cote rminal angles
-2
-4
Finding Coterminal Angles
Ex. Find and draw a positive angle and negative angle that are
coterminal with the given angle.
a)
b) 150
30
c)
2
3
radians
Evaluating Trig Functions Determined by a Point in Q1
Ex. Evaluate the six trig functions of the angle  , whose terminal side
contains the point (5, 3).
4
(5, 3)
2
34
3

-10
-5
5
-2
-4
5
10
10
Notes 4.3 pre-calc page 3
Definition: Trigonometric Functions of Any Angle
Let  be any angle in standard position and let P(x,y) be
any point on the terminal side of the angle (except the
origin). Let r denote the distance from P(x,y) to the
2
2
origin, r  x  y Then:
y
x
y
sin  
cos  
tan  
r
r
x
r
r
x
csc  
sec  
cot  
y
x
y
When considering the trig functions as functions of real
numbers, the angles will be measured in radians.
Definition: Unit Circle – the unit circle is a circle of
radius 1 centered at the origin.
(cos , sin )
Trig functions of real numbers
Let t be any real number, and let P(x,y) be the point
corresponding to t when the number line is wrapped onto
the unit circle. Then
y
sin t  y
cos t  x
tan t 
x
1
1
x
csc t 
sec t 
cot t 
y
x
y
Notes 4.3 pre-calc page 4
Therefore, the number t on the number line always wraps
onto the point (cos t, sin t) on the unit circle.
Definition – Periodic function – A function y = f(t) is
periodic if there is a positive number c such that
f(t +c) = f(t) for all values of t in the domain of f.
The smallest such number c is called the period of the
function.
Notes 4.4 pre-calc
Graphs of Sine and Cosine: Sinusoids
The period of a sine curve and cosine curve is 2
Definition of sinusoid – a function is a sinusoid if it can
be written in the form f(x) = a sin (bx+c) + d where a, b,
c, and d are constants and neither a nor b equals 0.
Any transformation of a sine function is a sinusoid.
Horizontal stretches and shrinks affect the period and the
frequency.
Vertical stretches and shrinks affect the amplitude
Horizontal translations bring about phase shifts.
Amplitude – the amplitude of the sinusoid
f(x) = a sin (bx+c) + d is a
Amplitude of f(x) = a cos (bx+c) + d is a
Graphically, the amplitude is half the height of the wave.
Example: 1 cos(x) ½ cos (x)
-3sin(x)
Period -- f(x) = a sin (bx+c) + d is
Period of f(x) = a cos (bx+c) + d is
2
b
2
b
Graphically the period is the length of one full cycle of
the wave.
Notes 4.4 pre-calc page 2
Example:
Find period y=sin(x)
x
y=-2sin( 3 ) y = 3 sin (-2x)
b
Frequency -- f(x) = a sin (bx+c) + d is 2
b
Frequency of f(x) = a cos (bx+c) + d is 2
Graphically the frequency is the number of complete
cycles the wave completes in a unit interval.
 2 x 
4sin


Example f(x) =
 3 
Period is 3
Amplitude is 4
Frequency is 1/3
Phase shift is “c”
Examples:

amplitude  6
Period is 5
f(x) = a sin (bx+c) + d
goes through  2, 0 
f(x) = 6 sin (10x) but this goes through (0,0)
0 = 6 sin (10(2) + c) so 10(2) + c = 0 so c = -20
Notes 4.4 pre-calc page 3
Graphs of f(x) = a sin (b(x-h) ) + k
f(x) = a cos (b(x-h) ) + k
where a  0 and b  0
amplitude = a
phase shift = h
vertical translation = k
period =
2
b
b
frequency = 2
Notes 4.5 precalc
Graphs of Tan, Cot, Sec, Csc
Tangent graph –
Domain – all reals except odd multiples of /2;
Range – all reals
Increasing; continuous (on domain);
symmetric with origin (odd)

vertical asymptote x  k  2 for all odd integers of k
lim tan x and lim tan x do not exist
x 
x 
Period = 
y  a tan(b( x  h))  k
a is (vertical stretch/shrink)
b is period
h is horizontal translation
k is vertical translation
Amplitude and phase shift are not used.
x cos t
cot angent  
y sin t
Asymptotes are at zeros of sine function. 0, 
Notes 4.5 pre-calc page 2
Secant has asymptotes at the zeros of the cosine function

3
2
and
2
Period is 2
Max = -1
min = 1 because cos  = 1 and sec  = 1
Cosecant asymptotes are at zeros of sin function 0, , 2
Period is 2
Max = -1
min = 1
Example: Solve graphically x2 = csc x
Let:
y1  x 2
y2  csc x
Learn table from page 400.
Notes 4.7 pre-calculus
Inverse Trig Functions
If domain is restricted of y = sin x to
  
  2 , 2 
the inverse
1
sine function y  sin x is the inverse of the restricted
sine function.
Inverse sine function (ArcSine function)
The unique angle y in the interval  2 , 2  such that


sin y = x is the inverse sine (or arcsine) of x denoted
sin 1 x
or arcSin x
1
The domain of y  sin x is [-1,1] and the range is
  
  2 , 2 
Inverse cosine function (ArcCosine function)
The unique angle y in the interval  o,   such that
cos y = x is the inverse cosine (or arccosine) of x denoted
cos 1 x
or arcCos x
1
The domain of y  cos x is [-1,1] and the range  o,  
Notes 4.7 pre-calculus page 2
The unique angle y in the interval
  
  2 , 2 
such that tan y=x
1
is the inverse tangent (or arctangent) of x denoted tan x
or arctan x
1
The domain y  tan x is (-,) and the range  2 , 2 


Always true:
sin(sin 1 x)  x
cos(cos1 x)  x
tan(tan 1 x)  x
True on restricted domain:
sin 1 (sin x)  x
cos1 (cos x)  x
tan 1 (tan x)  x
Notes 4.8 precalc
Solving Problems with Trigonometry
Angles of Elevation – the angle through which the eye
moves up from horizontal to look at something above.
Angle of Depression – angle through which the eye
moves down from horizontal to look at something below.
The sin and cos functions, because of periodic nature help
describe motion of objects that oscillate, vibrate or rotate.
Simple Harmonic Motion
A point moving on a number line is in simple harmonic
motion if its directed distance “d” from the origin is given
by either d=a sin t or d=a cos t where a and  are real

numbers and  > 0. The motion has frequency of 2 ,
which is the number of oscillations per unit of time.
Example 4 page 407
a is the radius of the wheel
d is the directed distance of the piston from its center of
oscillation
 is the rate the wheel rotates.
If r = 8 cm and = 8 rad/sec then d = 8 cos 8t The
8
frequency is 2  4