next up previous
Next: Complex Numbers Up: Basic Trigonometry Previous: Alternate Definitions of Sine

Definitions of Tangent, Secant, and Cosecant

 

Let tex2html_wrap_inline715 denote an angle.

Definition: The tangent of tex2html_wrap_inline715 , denoted tex2html_wrap_inline1015 is

displaymath999

Definition: The secant of tex2html_wrap_inline715 , denoted tex2html_wrap_inline1019 is

displaymath1000

Definition: The cosecant of tex2html_wrap_inline715 , denoted tex2html_wrap_inline1023 is

displaymath1001

displaymath1002

We summarize information about the domain and range of the six trig functions:

  1. tex2html_wrap_inline735 : Domain equals tex2html_wrap_inline1027 . Range equals [-1,1].
  2. tex2html_wrap_inline725 : Domain equals tex2html_wrap_inline1027 . Range equals [-1,1].
  3. tex2html_wrap_inline1015 : Domain equals tex2html_wrap_inline1039 . Range equals tex2html_wrap_inline1027 .
  4. tex2html_wrap_inline1043 : Domain equals tex2html_wrap_inline1045 . Range equals tex2html_wrap_inline1027 .
  5. tex2html_wrap_inline1019 : Domain equals tex2html_wrap_inline1039 . Range equals tex2html_wrap_inline1053 .
  6. tex2html_wrap_inline1023 : Domain equals tex2html_wrap_inline1045 . Range equals tex2html_wrap_inline1053 .

Using the alternate definition of sine and cosine as ratios of sides of right triangles, we can give alternate definitions of tangent, secant, cosecant, and cotangent.

  1. tex2html_wrap_inline1061 .
  2. tex2html_wrap_inline1063 .
  3. tex2html_wrap_inline1065 .
  4. tex2html_wrap_inline1067 .

Oscar Had A Heap Of Apples is a mnemonic to remember that tex2html_wrap_inline1069 , tex2html_wrap_inline1071 , and tex2html_wrap_inline1073 .

Starting with the known trigonometric identity tex2html_wrap_inline745 we can derive two more identities by dividing both sides of this equation by tex2html_wrap_inline1077 and by tex2html_wrap_inline1079 .

displaymath1003

displaymath1004



Carl Lee
Wed Apr 21 08:17:28 EDT 1999