Graphing Inverse Trigonometric Functions
Key Questions
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Throughout the following answer, I will assume that you are asking about trigonometry restricted to real numbers.
Using Domain of
arc sin xarcsinx Find
arc sin (3)arcsin(3) .33 is not in the domain ofarc sinarcsin , (33 is not in the range ofsinsin ) soarc sin (3)arcsin(3) does not exist.Using Range of
arc sin xarcsinx Find
arc sin (1/2)arcsin(12) .Although there are infinitely many
tt withsin t = 1/2sint=12 , the range ofarc sinarcsin restricts the value to thosett with(-pi)/2 <= t <= pi/2−π2≤t≤π2 , So the value we want ispi/6π6 .Using the quadrant
This is the same as using the range, but it involves thinking about the problem more geometrically.Find
arc sin (1/2)arcsin(12) .arc sin (1/2)arcsin(12) is a number (an angle) in quadrant I or IV. It is thett with smallest absolute value (the shortest path from the initial side).Again,
arc sin (1/2) = pi/6arcsin(12)=π6 . -
For a trig function, the range is called "Period"
For example, the function
f(x) = cos xf(x)=cosx has a period of2pi2π ; the functionf(x) = tan xf(x)=tanx has a period ofpiπ . Solving or graphing a trig function must cover a whole period.The range depends on each specific trig function.
For example, the inverse functionf(x) = 1/(cos x) = sec xf(x)=1cosx=secx has as period2pi2π .Its range varies from (+infinity) to Minimum
11 then back to (+infinity), between (-pi/2−π2 andpi/2π2 ).Its range also varies from (-infinity) to Max -1 then back to to (-infinity), between (
pi/2π2 and3pi/23π2 ). -
Since the graphs of
f(x)f(x) andf'(x) are symmetric about the liney=x , start with the graph of a trigonometric function with an appropriate restricted domain, then reflect it about the liney=x .(Caution: Their domains must be restricted to an appropriate interval so that their inverses exist.)
Let us sketch the graph of
y=sin^{-1}x .The graph of
y=sinx on[-pi/2, pi/2] looks like:By reflecting the graph above about the line
y=x ,The curve in purple is the graph of
y=sin^{-1}x .The graphs of other inverse trigonometric functions can be obtained similarly.
I hope that this was helpful.