Inverse Trigonometric Functions

1
Inverse Trigonometric Functions

• 6.1-6.2

2
Inverse Function 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.

3
Inverse Function Review

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

4
Inverse Function Review

• The domain of f is the range of f-1 and the range
  of f is the domain of f-1.
• The graph of f and f-1 are symmetric with respect
  to yx.
• f-1(f(x)) x for every x in the domain of f and
  f(f-1(x)) x for every x in the domain of f-1.

5
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.

6
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 .

7
Inverse Sine Function

• See graphs on page 448-449.

8
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.

9
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 .

10
Inverse Cosine Function

• See graphs on page 452-453.

11
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.

12
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).

13
Inverse Tangent Function

• See graphs on page 455-456.

14
Sin, Cos and Tan

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

15
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

16
Examples

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

17
Examples

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

18
Examples

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

19
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.

20
Examples

• Sin-1(sin( 2p/3))
• 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

21
Properties for Sin,Cos,Tan and their Inverses

• Sin-1(sin(x)) x has infinite solutions
• cos-1(cos(x)) x has infinite solutions
• tan-1(tan(x)) x has infinite solutions
• These following are true whenever they are
  defined.
• cos(cos-1(x)) one solution
• Sin(Sin-1(x)) one solution
• tan(tan-1(x)) one solution

22
F-1(F(x)) and F(F-1(x) )

• Suppose tan-1(tan x) x
• Range for tan-1 is (p/2,p/2)
• Therefore, the outer x needs to be in this
  interval
• Therefore the inner x needs to be in this
  interval, since x must equal x.

23
F-1(F(x)) and F(F-1(x) )

• Suppose tan(tan-1 x) x
• Domain for tan-1 is all reals
• therefore the x can equal any number.

24
F-1(F(x)) and F(F-1(x) )

• Suppose sin(sin-1(x)) x
• The domain for sin-1 is 1, 1.
• Therefore the inside x is in this interval.
• Therefore the outside x must do the same.

25
F-1(F(x)) and F(F-1(x) )

• Suppose sin-1(sin(x)) x
• The range for sin-1 is p/2, p/2.
• Therefore the outside x is in this interval.
• Therefore the inside x must do the same.

26
F-1(F(x)) and F(F-1(x) )

• Suppose cos-1(cos x) x
• The range for cos-1 is 0,p.
• Therefore outer x is in this interval.
• The inner x needs to equal the outer x, so the
  inside x is in this interval as well.

27
F-1(F(x)) and F(F-1(x) )

• Suppose cos(cos-1(x)) x
• The domain for cos-1 is 1, 1.
• Therefore the inside x must be in this interval.
• That means the outside x will be in this interval
  as well.

28
Examples

• Sin(Sin-1(1.8))
• There is no angle in the restricted domain or
  regular domain that will produce a number larger
  than 1 or smaller than -1. Therefore, the answer
  does not exist for this value.

29
ExamplesOuter function is not an inverse, inside
function is.

• 6.2--26
• Write the equation ? inside.
• Rewrite the inside so that it is no longer an
  inverse trig.
• Set up a triangle from the bowtie and label the
  sides from the ratio in the above equation.
• Use pythagorean theorem to find all three sides
  of the triangle.
• Rewrite the outside with the inside being
  replaced by ?.
• Write the answer as a ratio using the bowtie
  triangle and sohcahtoa.

30
ExamplesOuter function is not an inverse, inside
function is.

• 6.2--36
• Write the equation R inside.
• Find R by drawing the angle in standard position,
  and using allsintancos, bowtie triangle,
  sin,cos,tan,30,60,45.
• Rewrite the original expression with the inside
  being replaced by R.
• Evaluate the inverse trig of this ratio using
  sin,cos,tan,30,60,45, inverse trig ranges, and/or
  quadrantal info.