Transcript More Practice with Trigonometry

More Practice with
Trigonometry
Section 4.3b
Let’s consider…
Quadrantal Angle – angles whose terminal sides lie along
one of the coordinate axes
Note: Quadrantal angles do not produce reference
triangles, but we can still evaluate their trig. functions
P(0,1)
270
y 1
sin  270    1
r 1
More to consider…
Quadrantal Angle – angles whose terminal sides lie along
one of the coordinate axes
Note: Quadrantal angles do not produce reference
triangles, but we can still evaluate their trig. functions
3
P(–1,0)
y
0
0
tan3  
x 1
Still more to consider…
Quadrantal Angle – angles whose terminal sides lie along
one of the coordinate axes
Note: Quadrantal angles do not produce reference
triangles, but we can still evaluate their trig. functions
r 1
11
 
sec
2
x 0
Undefined!!!
11
2
P(0,–1)
Trigonometric Functions of Real
Numbers
Unit Circle – a circle with radius 1 centered at the origin
Wrapping Function – associates points on the number line
with points on the unit circle
When the line is “wrapped
P(x,y)
t > 0 around” the unit circle, each
point t will fall on a point of
t
the circle that lies on the
t
terminal side of an angle t
t
radians in standard position.
(1,0)
This allows for an even nicer
definition for trig functions…
Trigonometric 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
sin t  y
1
csc t   y  0 
y
P
1
cost  x sec t   x  0 
x
y
x
tan t   x  0  cot t   y  0 
y
x
t
t
Trigonometric 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
P  cos t ,sin t 
Note: The number t on the
number line always wraps
onto the point (cos t, sin t)
on the unit circle!!!
For this reason, the trig functions when
applied to real numbers are usually
called the circular functions.
t
t
Guided Practice
Find cos and tan  by using the given information to
construct a reference triangle.
3
1. sin  
7
and
tan   0
With sin > 0, we’re in
either QI or QII
With tan < 0, we have QII
Draw a reference triangle, with r = 7 and y = 3:

P 2 10,3
Use Pythag. Thm:
x  7  3  2 10
2
2

7
3
2 10
Guided Practice
Find cos and tan  by using the given information to
construct a reference triangle.
3
1. sin  
7
and
tan   0
Our Definitions:
2 10
cos   
7
3
tan   
2 10

P 2 10,3

7
3
2 10
Whiteboard Problems
Find cos and tan  by using the given information to
construct a reference triangle.
With sec > 0, we’re in
2. sec  3 and sin   0 either QI or QIV
With sin > 0, we have QI

P 1, 2 2
3
2 2
1

Our Definitions:
1
cos  
3
tan   2 2
Whiteboard Problems
Find cos and tan  by using the given information to
construct a reference triangle.
With cot und., we
3. cot  is und. and sec  0
have y = 0
With sec < 0, terminal
side is on neg. x-axis
P  1,0 
1
Our Definitions:
cos  1
0
tan  
0
1
Whiteboard Problems
1
1. Find cos and cot  if sin   and tan   0.
4
3
2. Find sin and cos if cot  and sec  0.
7
4
3. Find csc and cot  if tan   and sin  0.
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