What are the reciprocal trig functions?

• What do the graphs of the reciprocal trig functions look like?

• The graph of?y?=?secx?has the following properties:
  • The?y-axis?is a?line of symmetry
  • It has a?period?of?360°?(2π?radians)
  • There are vertical?asymptotes?wherever?cos?x?=?0
    • If drawing the graph without the help of a GDC it is a good idea to sketch cos?x?first and draw these in
  • The domain?is all?x?except?odd multiples of 90°?(90°, -90°, 270°, -270°, etc.)
    • in?radians?this is all?x?except odd multiples of π/2?(π/2, - π/2, 3π/2, -3π/2, etc.)
  • The range?is?y?≤?-1?or?y?≥?1

• The graph of?y?= cosec?x?has the following properties:
  • It has a?period?of?360°?(2π?radians)
  • There are vertical?asymptotes?wherever?sin?x?=?0
    • If drawing the graph it is a good idea to sketch sin?x?first and draw these in
  • The?domain?is all?x?except multiples of 180°?(0°, 180°, -180°, 360°, -360°, etc.)
    • in radians this is all?x?except multiples of π?(0, π, - π, 2π, -2π, etc.)
  • The?range is?y?≤?-1?or?y?≥?1

• ?The graph of?y?= cot?x?has the following properties
  • It has a?period?of?180°?or?π?radians
  • There are vertical?asymptotes?wherever?tan?x?=?0
  • The?domain?is all?x?except multiples of 180°?(0°, 180°, -180°, 360°, -360°, etc.)
    • In?radians this is all?x?except multiples of π?(0, π, - π, 2π, -2π, etc.)
  • The?range?is?y?∈???(i.e. cot can take?any?real number value)

What are the Pythagorean Identities?

• Aside from the Pythagorean identity sin²x?+?cos²x?= 1there are two further Pythagorean identities you will need to learn
  • Both can be found in the formula booklet
• Both of these identities can be derived from sin²x?+?cos²x?= 1
  • To derive the identity for?sec²x?divide?sin²x?+?cos²x?= 1by?cos²x
  • To derive the identity for?cosec²x?divide?sin²x?+?cos²x?= 1by?sin²x