4-7 Inverse Trig Functions

8/3/2019 4-7 Inverse Trig Functions

1/17

1

Trig/Precalc

Chapter 4.7 Inverse trigfunctions

Objectives

Evaluate and graph the inverse

sine functionEvaluate and graph the remaining

five inverse trig functions

Evaluate and graph thecomposition of trig functions


2/17

2

The basic sine function fails the horizontal line test.

It is not one-to-one so we cant find an inverse

function unless we restrict the domain.

Highlight the curve/2 < x < /2

On the interval [-/2, /2]for sin x:the domain is [-/2, /2]and the range is [-1, 1]

We switch x and y to get inverse functionsSo for f(x) = sin-1 xthe domain is [-1, 1] andrange is [-/2, /2]

2/2-/2

y = sin(x)

Therefore


3/17

3

Graphing the Inverse

When we get rid of all theduplicate numbers we getthis curve

Next we rotate it across they=x line producing this curve

-10 -5 5 10

-5

5

First we draw the sin curve

This gives us:Domain : [-1 , 1]

Range: 2, 2


4/17

4

Inverse sine function

y = sin-1 x or y = arcsin x The sine function gives us

ratios representing oppositeover hypotenuse in all 4quadrants.

The inverse sine gives us theangle or arc length on the unitcircle that has the given ratio.

Remember the phrase arcsine of x is theangle or arc whose sine is x.

/2

-/2

1


5/17

5

Evaluating Inverse Sine

If possible, find the exact value.

a. arcsin(-1/2) = ____

We need to find the angle in the range

[-/2, /2] such that sin y = -1/2

What angle has a sin of ? _______What quadrant would it be negative and within

the range of arcsin? ____

Therefore the angle would be ______

6

IV

6

6


6/17

6

Evaluating Inverse Sine cont.

b. sin-1( ) = ____

We need to find the angle in the range [-/2, /2] such thatsin y =

What angle has a sin of ? _______

What quadrant would it be positive and within the range ofarcsin? ____


c. sin-1(2) = _________Sin domain is [-1, 1], therefore No solution

3

3

2

3

2

3

2

3 2

1

I

3

3

No Solution


7/17

7

Graphs of Inverse

Trigonometric Functions

The basic idea of the arc function is the same

whether it is arcsin, arccos, or arctan


8/17

8

Inverse Functions Domains and

Ranges y = arcsin x

Domain: [-1, 1]

Range:

y = arccos x

Domain: [ -1, 1]

Range:

y = arctan x

Domain: (-, )

Range:

,2 2

0,

,2 2

y = Arcsin (x)

y = Arccos (x)

y = Arctan (x)


9/17

9

Evaluating Inverse Cosine

If possible, find the exact value.

a. arccos((2)/2) = ____We need to find the angle in the range

[0, ] such that cos y = (2)/2

What angle has a cos of (2)/2 ? _______

What quadrant would it be positive and within the range of arccos? ____


b. cos-1(-1) = __What angle has a cos of -1 ? _______


10/17

10

Warnings and Cautions!

Inverse trig functions are equal to the arc trigfunction. Ex: sin-1 = arcsin

Inverse trig functions are NOT equal to the

reciprocal of the trig function.Ex: sin-1 1/sin

There are NO calculator keys for: sec-1 x, csc-1 x,

or cot-1

x

And csc-1x 1/csc xsec-1x 1/sec x

cot-1

x 1/cot x


11/17

11

Evaluating Inverse functions

with calculators ([E] 25 & 34)

If possible, approximate

to 2 decimal places.

19. arccos(0.28) = ____

22. arctan(15) = _____

26. cos-1(0.26) = ____

34. tan-1(-95/7) = ____

Use radian mode unlessdegrees are asked for.


12/17

12

Guided practice

Example of [E] 28 & 30

Use an inverse trig function

to write as a function of x.

28. Cos = 4/x so

= cos-1(4/x) where x > 0

30. tan = (x 1)/(x2 1) = tan-1(x 1)/(x2 1)

where x 1 > 0 , x > 1

as a function of xmeans to write an equationof the form equal to anexpression with x in it.

4

x

1

10

x


13/17

13

Composition of trig functions

Find the exact value, sketch a triangle.

cos(tan-1 (2)) = _____

This means tan = 2 sodraw the triangle

Label the adjacent and opposite sides

Find the hypo. using Pyth. Theorem

So the

2

1

5

2 5cos

5


14/17

14

Example

Write an algebraic expression that is equivalent tothe given expression.

cot(arctan(1/x))

u

x

1

2cot

1

xu

x

1) Draw and label the triangle

---(let u be the unknown angle)

2) Use the Pyth. Theo. to compute the hypo

3) Find the cot of u

21x


15/17

You Try!

Evaluate:

-4/3

0 rad.

csc[arccos(-2/3)] (Hint: Draw a triangle)

Rewrite as an algebraic expression:

3arcsin

2

3arcsin sin

2

3

tan arccos5

arccos tan 2

3

2

3 5 5

2

2

1

1

v

v

A


16/17

Word problem involving sin or cos function:

P type 1

pcalc643

ALEKS

An object moves in simple harmonic motion with amplitude 12 cmand period 0.1 seconds. At time t = 0 seconds , its displacementdfrom rest is 12 in a negative direction, and initially it moves ina negative direction.

Give the equation modeling the displacement das a function of

time t.

Undo HelpClear

Next >> Explain

A


17/17

Word problem involving sin or cos function:

P type 2

pcalc643

ALEKS

The depth of the water in a bay varies throughout the day with the tides.Suppose that we can model the depth of the water with the followingfunction. h(t) = 13 + 6.5 sin 0.25t

In this equation, h(t) is the depth of the water in feet, and tis the time inhours.

Find the following. If necessary, round to the nearest hundredth.

Frequency of h: cycles per hourPeriod of h: hoursMinimum depth of the water: feet Undo HelpClear

Next >> Explain

Documents

4-7 Inverse Trig Functions