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Lesson 4.7. Inverse Trigonometric Functions.Previously you have
learned To find an inverse of a function, let every x be y and
every y be x, then solve the equation for y. Inverse function
notation f(x) For a function to have an inverse it has to be
one-to-one. One x for one y value, and one y for one x value. It
will pass the vertical and the horizontal line test. Two inverse
functions on the graph reflect over y=x f(x) f(x) (x,y) (y, x)
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Sometimes you just dont have a nice or convenientalgebraic
process that will give you an inverse function. Many functions need
a special, new rule for their inverse. Some examples of these
functions are:
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FINDING INVERSE OF A TRIG FUNCTION :Given f(x) = sin (x) y = sin
(x) change f(x) for y x = sin (y) switch xs and ys y = arcsin (x)
solve for y f(x) = arcsin (x) write using function notation f(x) =
sin(x)
Inverse Trigonometric function notation: Inverse sine
Inverse cosine
Inverse tangent
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FINDING INVERSE OF A TRIG FUNCTION ALGEBRAICALLY :
Given y = sin (x) - = sin ( -/6) sin (-/6) = - (-/6) = sin (- )
switch x and y values (-/6) = arcsin ( - ) solve for y
(-/6) = sin( - ) sin( - ) = (-/6)
How does this look on the graph ??
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Graphing Inverse Trigonometric functions:The graph of y = sin
x
D: all reals R: [-1,1]Period: 2 Y-int.(0,0)
this function is not one-to-one (different x values yield the
same y)with domain restricted to [-/2 ; /2] y = sin x is one-to
onetherefore, we can use this piece to finds its inverse
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Using the domain restricted to [-/2 ; /2] lets graph y = arcsin
x Remember that y = arcsin x is equivalent to sin y = x
Xy-values of sin xy = arcsin xx values of sin x
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Graphs of inverse functions The graph of y = arc sin x Domain:
Range:
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NEXT ON THE LIST
ARCCOSINE Sketch the common curve of y = cos xSelect interval of
the cosine that will be one-to-oneSketch the graph of arccosine
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The chosen section for the cosine is in the red frame. This
section includes all outputs from 1 to 1 and all inputs in the
first and second quadrants.
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Graphs of inverse functions The graph of y = arccos x Domain:
Range:
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The other trig functions require similar restrictions on their
domains in order to generate an inverse.y=tan(x)y=arctan(x)
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Graphs of inverse functions The graph of y = arctan x Domain:
Range:
- The table below will summarize the parameters we have so far.
Remember, the angle is the input for a trig function and the ratio
is the output. For the inverse trig functions the ratio is the
input and the angle is the output.When x
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Evaluating Inverse Trigonometric Functions algebraically.
When evaluating inverse trigonometric functions, you are looking
for the angle whose (insert: sin, cos, or tan) is x. Also, keep in
mind the domain and range of each function.
Find the exact value for
Solution: For what value of x is sin (x) = ? Sin ( /3) =
therefore
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Evaluating Inverse Trigonometric Functions algebraically.
Find the exact value for
Solution: For what value of x is cos (x) = ?
Cos ( 3/4) = ; therefore