Download 4.3 Trigonometry Extended: The Circular Functions

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 with the ground. 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
SOH CAH TOA - Redefined
The Big 3
Trig Ftns
Old School
New School
Some Space for the Menorah
sin 
cos 
tan 
cot 
sec 
csc 
Reciprocal
Ftns
Old School
New School
Evaluating Trig Functions Based on Location
Ex 2: 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 3:
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) .
Key-Points
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 4: Where is  ?
sin   0
tan   0
Key-Points
cot   0
sin   0
sec   0
csc   0
THE UNIT CIRCLE
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 )  cos 2 (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,  , is drawn
through the point on the unit circle given by:
2 2 1
,  

3
3

Key-Points
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