Transcript Reciprocal functions secant, cosecant, cotangent

Reciprocal functions
secant, cosecant, cotangent
secant  sec
cosecant  csc
cotangent  cot
Secant is the reciprocal of cosine.
Reciprocal means to flip the ratio.
Adj
cos 
Hyp
Cosecant is the reciprocal of sine.
Reciprocal means to flip the ratio.
Opp
Hyp
sin  
csc 
Opp
Hyp
Hyp
sec 
Adj
Cotangent is the reciprocal of
tangent. Reciprocal means to flip
the ratio.
Opp
tan 
Adj
Adj
cot 
Opp
Sec, csc & cot also have the same SIGN as their reciprocals.
S
Just sin
and csc
positive
T
Just tan
and cot
positive
A
Now all six
functions
are positive
C
Just cos
and sec
positive
Determine the quadrant in which x terminates:
1. sec x  0 and tan  0
sec x  0 meansnegative
Since sec is the reciprocal function of cos,
then sec and cos have the same SIGN in
the quadrants. If sec is negative, so is cos.
Thus we are in quadrants II and III.
Tangent is positive, so we are in quadrants
I and III.
x
S
Just sin and
csc positive
xx
T
Just tan
and cot
positive
x
A
Now all six
functions are
positive
C
Just cos
and sec
positive
Since they overlap in
quadrant III, then that is
where x terminates!
Determine the quadrant in which x terminates:
2. csc x  0 and cot x  0
xx S
Just sin and
csc positive
T
Think,
where is
sin
positive?
Think,
where is
tan
negative?
Just tan
and cot
positive
x
A
Now all six
functions are
positive
x
C
Just cos
and sec
positive
Since they overlap in
quadrant II, then that is
where x terminates!
Determine the quadrant in which x terminates:
3. sin x  0 and sec x  0
x
S
A
Just sin and
csc positive
Now all six
functions are
positive
xx
T
Think,
where is
sin
negative?
Think,
where is
cos
negative?
Just tan
and cot
positive
x
C
Just cos
and sec
positive
Since they overlap in
quadrant III, then that is
where x terminates!
Find the exact value of each expression:
Step:
sec 120 
60 
1. Start by drawing the given angle
120 
3. Rewrite the function using reference
angle
180  120  60
sec 60 
cos 60
1
 cos 60  
2
2. Find the reference angle
4. Determine SIGN of function in
quadrant where it was drawn
5.
Use cos because
sec is not on exact
value chart and cos
is reciprocal of sec!
This is how you can check
Now find exact value using exact
your chart
answer
the function to
value
(use on
reciprocal
get
value off chart)
calculator!
6. Take reciprocal of value off chart to
get final answer!
Since cos is negative in II, sec is as well
Cos in quad II is negative
 sec 60   2
When you switch
back to sec, you
take reciprocal of
exact value!
Find the exact value of each expression:
cot 300 
3
1
3
3




3
9
3 3
300 
60 
360  300  60
cot 60 
tan 60 
 tan 60    3
1
 cot 60   
3
We have a
problem here!
3
 cot 60   
3
Find the exact value of each expression:
2
 csc 45   
2
csc 225 
2
2
2 2
2 2




2 2
2
4
225 
45 
225  180  45
 2
 csc 45    2
csc 45 
sin 45 
 sin 45   
2
2
Page 10
sec 150 
30 
150 
sec 30cos30
 sec 30 cos30

3
 sec 30  
 2 
Sec is the reciprocal of cos, so:
180  150  30
3  1
 2 


 
3  2 

Page 10
csc 238   1.1791
csc 238   1.179
Page 10
cot 35  1.4281
cot 35  1.428
Page 10
3
cos30 
2
2
sec 30 
3
f x   2 sec x
f 30  2 sec 30
2
2 sec 30  2
3
4
2 sec 30 
3
4 3
2 sec 30 
3
4
3

3 3
4 3
9
4 3
3