IB DP Maths: AA HL复习笔记3.7.1 Reciprocal Trig Functions

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 ∈ ℝ (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