What are the inverse trig functions?

• The functions?arcsin,?arccos?and?arctan?are the?inverse functions?of?sin,?cos?and?tan?respectively when their domains are restricted
  • sin (arcsin?x) =?x??for? -1 ≤?x?≤ 1
  • cos (arccos?x)?=?x??for? -1 ≤?x?≤ 1
  • tan (arctan?x) =?x??for all?x
• You will have seen and used the inverse trig?operations?many times already
  • Arcsin is the operation?sin^-1?^?
  • Arccos is to the operation?cos^-1?
  • Arctan is the operation?tan^-1
• The domains of?sin,?cos, and?tan?must first be restricted to make them?one-to-one functions
  • A function can only have an inverse if it is a one-to-one function
• The domain of?sin?x?is restricted to?-π/2?≤?x?≤?π/2??(-90°?≤?x?≤?90°)
• The domain of?cos?x?is restricted to?0?≤?x?≤?π??(0°?≤?x?≤?180°)
• The domain of?tan?x?is restricted to?-π/2?<?x?<?π/2??(-90°?<?x?<?90°)
• Be aware that sin^-1?x, cos^-1?x, and tan^-1?x?are?not?the same as the reciprocal trig functions
  • They are used to solve trig equations such as sin?x?= 0.5 for all values of?x
  • arcsin?x?is the same as sin^-1?x?but not the same as (sin?x)^-1

What do the graphs of the inverse trig functions look like?

• The graphs of?arcsin,?arccos?and?arctan?are the?reflections?of the graphs of?sin,?cos?and?tan?(after their domains have been restricted) in the line?y?=?x
  • The?domains?of arcsin?x?and arccos?x?are both -1 ≤?x?≤ 1
  • The?range?of arcsin?x?is -π/2 ≤?y?≤ π/2

• • The?range?of arccos?x?is 0 ≤?y?≤ π

• The?domain?of arctan?x?is?x?∈ ?
• The?range?of arctan?x?is -π/2 <?y?< π/2
  • Note that there are horizontal asymptotes at π/2 and -π/2

How are the inverse trig functions used?

• The functions?arcsin,?arccos?and?arctan?are used to evaluate trigonometric equations such as sin?x?= 0.5
  • If sin?x?= 0.5 then arcsin?0.5 =?x?for values of?x?between -π/2 ≤?x?≤ π/2
    • You can then use symmetries of the trig function to find solutions over other intervals
• The inverse trig functions are also used to help evaluate algebraic expressions
  • From sin (arcsin?x) =?x?we can?also say that sinⁿ(arcsin?x) =?x^n???for? -1 ≤?x?≤ 1
  • If using an inverse trig function to evaluate an algebraic expression then remember to consider the domain and range of the function
    • arcsin(sin?x) =?x??only for??-π/2 ≤?x?≤ π/2
    • arccos(cos?x) =?x??only for??0 ≤?x?≤ π
    • arctan(tan?x) =?x??only for??-π/2 <?x?< π/2
  • The symmetries of the trig functions can be used when values lie outside of the domain or range
    • Using sin(x) = sin(π -?x) you get arcsin(sin(2π/3)) = arcsin(sin(π/3)) = π/3