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Inverse Trigonometric Functions

- 6.1-6.2

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.

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.

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.

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 .

Inverse Sine Function Graph

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 .

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.

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.

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.

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 .

Inverse Cosine Function Graph

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

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.

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).

Inverse Tangent Function Graph

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

Examples

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

Examples

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

Examples

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

Sin, Cos and Tan

- We used restricted domains of these functions

just to get the inverses. The functions

themselves keep their original domains.

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.

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.

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

Inverse Sine Example 4

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

is p/9.

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.

Example 6

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

value.

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

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.

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).

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.

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.

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

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

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

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

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