Inverse Trigonometric Functions

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Inverse Trigonometric Functions

• 6.1-6.2

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4.1-4.2 Review

• Each function has an inverse relation. The
  inverse relation is a function only when the
  original function is
• 1-1
• Some functions are important enough that we want
  to study their inverse behavior despite the fact
  that they are not 1-1. We do this by
• Restricting the domain.

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4.1-4.2 Review

• We restrict it to an interval on which it is 1-1,
  then we find the inverse of the restricted
  function.

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Inverse Sine Function

• The unique angle y in the interval -p/2, p/2
  such that sin y x is the inverse sine or
  arcsine of x, denoted sin-1 x or arcsin x.

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Inverse Sine Function

• So sin-1 x y where y is an angle.
• Domain of ysin-1 x is
• -1,1 and the range is
• -p/2, p/2 .

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Inverse Sine Function Graph
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Inverse Sine Example 1

• Arcsin(1/2) ?
• What angle has a sin of ½?
• The range for arcsin puts us in quadrant 1 or 4.
  Sin is positive only in quadrant 1, so that is
  where our answer lies. Use a table that gives the
  sin, cos, tan of 30,60,45 to find this out. 30
  degrees or p/6 has a sine of ½, so the answer is
  p/6 .

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Inverse Sine Example 2

• Arcsin(-sqrt(3)/2) ?
• What angle has a sin of sqrt(3)/2?
• The range for arcsin puts us in quadrant 1 or 4.
  Sin is negative only in quadrant 4, so that is
  where our answer lies. 60 degrees or p/3 has a
  sine of sqrt(3)/2, so the answer is the p/3 in
  quadrant 4, which is 5p/3.

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Inverse Sine Example 3

• Arcsin(p/2) ?
• What angle has a sin of p/2 ?
• The domain for arcsin is between 1 and -1, p/2 is
  more than one, so there is no possible answer for
  this.

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Inverse Cosine Function

• The unique angle y in the interval 0, p such
  that cos y x is the inverse cosine or arccosine
  of x, denoted cos-1 x or arccos x.

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Inverse Cosine Function

• So cos-1 x y where y is an angle.
• Domain of ycos-1 x is
• -1,1 and the range is
• 0, p .

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Inverse Cosine Function Graph
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Examples

• Cos-1(1/2) ?
• Use a table that gives the sin, cos, tan of
  30,60,45 to find this out.
• The angle whose cosine is ½ is 60 degrees.
• Cos-1(1/2) 60 degrees or pi/3

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Inverse Tangent Function

• The unique angle y in the interval (-p/2, p/2)
  such that tan y x is the inverse tangent or
  arctangent of x, denoted tan-1 x or arctan x.

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Inverse Tangent Function

• So tan-1 x y where y is an angle.
• Domain of ytan-1 x is
• (-8, 8) and the range is
• (-p/2, p/2).

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Inverse Tangent Function Graph
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Examples

• tan-1(1) ?
• Use a table that gives the sin, cos, tan of
  30,60,45 to find this out.
• The angle whose tangent is 1 is 45 degrees.
• tan-1(1) 45 degrees or pi/4

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Examples

• Cos-1(1) ?
• Cos is the x value.
• X is 1 at 0 radians
• Or 0 degrees.
• Cos-1(1) 0

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Examples

• Cos-1(-1) ?
• Cos is the x value.
• X is -1 at pi radians
• Or 180 degrees.
• Cos-1(-1) 180
• or pi

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Examples

• Cos is the x value
• Sin is the y value
• Tan is y/x

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Sin, Cos and Tan

• We used restricted domains of these functions
  just to get the inverses. The functions
  themselves keep their original domains.

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Properties for Sin,Cos,Tan and their Inverses

• Sin-1(sin(x)) x where p/2 x p/2
• True because x lies in the range for arcsin.
• If x is not in the range, then you have to
  calculate the sin(x) and then the arcsin of the
  answer.
• Sin(Sin-1(x)) x where -1 x 1
• True because x lies in the domain for arcsin.
• If x is not in arcsins domain, then it is
  unsolvable.

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Properties for Sin,Cos,Tan and their Inverses

• cos-1(cos(x)) x where 0 x p
• True because x is in the range of arccos.
• If x is not in the range, then you have to
  calculate the cos(x) and then the arccos of the
  answer.
• cos(cos-1(x)) x where -1 x 1
• True because x lies in the domain for arccos.
• If x is not in arccoss domain, then it is
  unsolvable.

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Properties for Sin,Cos,Tan and their Inverses

• tan-1(tan(x)) x where p/2 ltxlt p/2
• True because x is in the range of arctan.
• If x is not in the range, then you have to
  calculate the tan(x) and then the arctan of the
  answer.
• tan(tan-1(x)) x where -8 ltxlt 8

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Inverse Sine Example 4

• Arcsin(sin(p/9)) ?
• Since p/9 is in the range for arcsin, then answer
  is p/9.

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Inverse Sine Example 5

• Arcsin(sin(5p/6)) ?
• Since 5p/6 is not in the range for arcsin, we
  need to find the sin(5p/6).
• Sin(p/6) ½, so sin(5p/6) ½
• Now find arcsin(1/2).
• What angle has a sin of ½?
• p/6 does and this is in arcsins range.
• p/6 is the answer.

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Example 6

• Sin(Sin-1(1.8))
• 1.8 is outside of arcsins domain.
• Therefore, the answer does not exist for this
  value.

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Example 7

• Sin-1(sin(2p/3))
• 2p/3 is not in arcsins range, so..
• Find the sin(2p/3)
• sin(2p/3) sqrt(3)/2
• Then find the inverse sin of the result.
• Arcsin(sqrt(3)/2) p/3.
• Since sin(p/3) sqrt(3)/2

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Evaluating Composites of Trigonometric Functions

• Evaluate the inside first. If the inside is an
  inverse function, then watch the range. If the
  inside is a regular trig function, then use the
  bowtie triangle, drawing the angle in standard
  position, allsintancos, and the unit circle to
  find the value.
• Use the answer to the inside as the input for the
  outside function. If the outside is an inverse
  function, then watch the range.

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Evaluating Composites of Trigonometric Functions

• Find the Arccos(sin(7pi/6)).
• Sin(7pi/6) sin(210degrees)
• Sin(30) ½ so by allsintancos
• Sin(210degrees) -1/2
• Now we need to find arccos(-1/2).

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Evaluating Composites of Trigonometric Functions

• Arccos(-1/2) ? therefore,
• Cos(?) -1/2
• The range for arccos is 0,pi, so ? must fit in
  this interval.
• T must be in quadrant 2 since cosine is negative.
  
• The angle must be a multiple of pi/3, since
  cos(pi/3)1/2, so ? is 2pi/3.

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Find an Algebraic Expression Equivalent to the
Given Expression

• Create a right triangle with one of the acute
  angles being ?.
• Set the inside inverse function to theta.
• Rewrite as the corresponding function.
• Use this function to set up two of the sides of
  the triangle.
• Solve for the third side with pythagorean
  theorem.
• Finally evaluate the outside function at theta
  using the created triangle.

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Example 1

• Cos(arctan(x))
• Arctan(x) ? so tan(?) x
• Opposite side is x
• Adjacent side is 1
• Hyp is sqrt(x2 1)
• Cos(?) 1/ sqrt(x2 1)

• sqrt(x2 1)

X
?
1
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Example 2

• Cot(arccos(x))
• Arccos(x) ? so cos(?) x
• Adjacent side is x
• Hyp is 1
• Opposite side is sqrt(1-x2)
• Cot(?) x/ sqrt(1-x2)

• 1

Sqrt(1-x2)
?
x
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Example 3

• sin(arccos(3x))
• Arccos(3x) ? so cos(?) 3x
• Adjacent side is 3x
• Hyp is 1
• Opposite side is sqrt(1-9x2)
• sin(?) sqrt(1-9x2)

• 1

Sqrt(1-9x2)
?
3x
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Example 4

• tan(arccos(1/3))
• Arccos(1/3) ? so cos(?) 1/3
• Adjacent side is 1
• Hyp is 3
• Opposite side is sqrt(8)
• tan(?) sqrt(8)/1

• 3

Sqrt(8)
?
1