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4.3 Trigonometry Extended: The Circular
Functions
Objective:



Find the trig function value of any angle.
Explore trig functions as periodic functions.
Explore and use the unit circle.
WARMUP
(1) The value of csc  
5
. Find the values of the other five trig functions.
2
(2) A 12-foot ladder leans against the side of a house at an angle of 72 . How high up
the house is the ladder?
Drawing an Angle
Coterminal Angles
Coterminal angles are angles that _____________________________. There are
an infinite number of angles that fit this description. For instance, 70 is coterminal with
430 AND 290 .
Ex 1: Find one positive and one negative
coterminal angle for each of the following angle
measures:
305
Key-Points
132
Understanding Signs
The signs of an ordered pair, coupled with our trig function definitions hold valuable
clues that can aid us in evaluating trig functions.
Ex 2: Where is  ?
sin   0
tan   0
cot   0
sin   0
Key-Points
Evaluating Trig Functions Based on Location
sec   0
csc   0
Ex 3: Find the values of all 6 trig functions if an
angle,  , is drawn through the point given.
Key-Points
( 8, 6)
(2 3, 2)
Ex 4:
Find the exact value of sec if the terminal side of
an angle passes through the point (24, 7) .
Find the exact value of sin  if the terminal side of
an angle passes through the point (  3, 4) .
THE UNIT CIRCLE
Key-Points
Exploring the Unit Circle – With your group….
(1) For any value of t, the value of cos(t ) lies between _____ and _____ inclusive.
(2) For any value of t, the value of sin(t ) lies between _____ and _____ inclusive.
(3) The values of cos(t ) and cos(t ) are ___________________ each other,
making this function even.
(4) The values of sin(t ) and sin(t ) are ___________________ each other,
making this function odd.
(5) The values of sin(t ) and sin(t  2 ) are ___________________ each other. In
fact, this is true of all of the six trig functions on their domains.
(6) The values of sin(t ) and sin(t   ) are ___________________ each other. The
same is true for cos(t ) and cos(t   ) .
(7) The values of tan(t ) and tan(t   ) are ___________________ each other,
unless they are both undefined.
(8) The sum sin 2 (t )  cos2 (t ) is always equal to _____.
Ex 5: Use your knowledge of the unit circle to evaluate the following:
cos 225
sec 315
cot 5
tan 690
csc(420 )
Ex 6: Find the values of all 6 trig functions if an angle,  , Key-Points
is drawn through the point on the unit circle given by:
 2 2 1
,  

3
3

Using One Trig Ratio to Find the Others
5

Ex 7: Find the exact value of tan  if csc    and    
2
2
.
1
Find the exact value of cot  if cos    and csc  0 .
3
Find the exact value of cot  if sec  5 and sin   0 .
K-Points
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