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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
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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
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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
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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
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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
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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? ____
Therefore the angle would be ______
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
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Graphs of Inverse
Trigonometric Functions
The basic idea of the arc function is the same
whether it is arcsin, arccos, or arctan
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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)
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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? ____
Therefore the angle would be ______
b. cos-1(-1) = __What angle has a cos of -1 ? _______
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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
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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.
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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
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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
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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
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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
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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
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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
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