Honors Geometry
Advanced Trigonometry
9. The Other Three Trig Functions
Definitions of the new functions
We defined the cosine function as
x
cos =
r
We now define the secant function as
r
1
sec =
=
x
cos 
We defined the sine function as
y
sin =
r
We now define the cosecant function as
1
r
csc =
=
sin 
y
We defined the tangent function as
y
sin 
tan =
=
cos 
x
We now define the cotangent function as
cot  =
1
cos 
x
=
=
tan 
sin 
y
Range of values for the new functions
Since -1 < cos < 1,
sec < -1 or sec > 1
Since -1 < sin < 1,
csc < -1 or csc > 1
Since tangent can equal any number, its
reciprocal, cotangent can equal any
number.
Finding values of the new functions
There are no secant, cosecant, or tangent
buttons on our calculators. To find
specific values of these functions, we
use the fact that they are reciprocals of
our previous functions.
1
cos 20
1
=
0.9396926208
= 1.064177772
= 1.0642
sec 20 =
You can find cot 146 in one operation
on your calculator by entering
1/tan 146
You will get –1.482560969, or –1.4826
Finding inverse values
How would we find the value of
csc-1(4)
When we are computing an inverse trig
function, we are finding the measure of
an angle. Lets call that angle .
csc-1(4) = 
This means that
csc  = 4
or
1
4
=
1
sin 
Flipping both sides over,
sin = ¼
Expressing as inverse sine
 = sin-1(1/4)
Since csc-1(4) =  and  = sin-1(1/4),
then csc-1(4) = sin-1(1/4) hmm…
= 14.5
We notice that 4 and ¼ are reciprocals of
each other. This pattern always holds.
sec-1(4/3) = cos-1(3/4) = 41.4
csc-1(2.5) = sin-1(1/2.5) = 23.6
cot-1(7) = tan-1(1/7) = 8.1
sec-1(-2) = cos-1(1/-2) = 120
cot-1(-4.7) = tan-1(1/-4.7) = -12 = 348
csc-1(-1) = cos-1(1/-1) = 180
Honors Geometry
Advanced Trigonometry
9. The Other Three Trig Functions
Find the quantity.
16. sec-1(3)
1. sec 35
17. csc-1(7)
2. csc 28
18. cot-1(4)
3. cot 79
19. sec-1(3.5)
4. sec 143
20. csc-1(5.1)
5. csc 266
21. cot-1(0.2)
6. cot 216
22. sec-1(-2)
7. sec 56.2
8. csc 83.1
23. csc-1(-5)
24. cot-1(-0.25)
25. sec-1(-2.5)
9. cot 111.8
26. csc-1(-4.1)
10. sec -100
27. cot-1(-1)
11. csc -82
28. sec-1( 2 )
12. cot -200
13. sec –127.2
14. csc -357
15. cot -135
29. csc-1(2)
30. cot-1( 3 )
2
31. sec-1  
 3
32. csc-1(1)
 3
33. cot-1  3 

