1 trig/precalc chapter 4.7 inverse trig functions objectives evaluate and graph the inverse sine...
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Trig/PrecalcChapter 4.7 Inverse trig functions
ObjectivesEvaluate and graph the inverse
sine functionEvaluate and graph the remaining
five inverse trig functionsEvaluate and graph the
composition of trig functions
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The basic sine function fails the horizontal line test. It is not one-to-one so we can’t find an inverse function unless we restrict the domain. Highlight the curve –π/2 < x < π/2
On the interval [-π/2, π/2] for sin x: the domain is [-π/2, π/2] and the range is [-1, 1]
We switch x and y to get inverse functions So for f(x) = sin-1 x the domain is [-1, 1] and range is [-π/2, π/2]
π 2ππ/2-π/2
y = sin(x)
Therefore
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Graphing the Inverse
When we get rid of all the duplicate numbers we get this curve
Next we rotate it across the y=x line producing this curve
-10 -5 5 10
6
4
2
-2
-4
-6
-10
-55
10
6 4 2 -2 -4 -6
First we draw the sin curve
This gives us:Domain : [-1 , 1]
Range: 2, 2
4
4
2
-2
-4
-5 5
Inverse sine function y = sin-1 x or y = arcsin x
The sine function gives us ratios representing opposite over hypotenuse in all 4 quadrants.
The inverse sine gives us the angle or arc length on the unit circle that has the given ratio.
Remember the phrase “arcsine of x is the angle or arc whose sine is x”.
π/2
-π/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 [-π/2, π/2] such that sin y = -1/2
What angle has a sin of ½? _______What quadrant would it be negative and within
the range of arcsin? ____Therefore the angle would be ______
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IV6
6
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Evaluating Inverse Sine cont.
b. sin-1( ) = ____ We need to find the angle in the range [-π/2, π/2] such that
sin y =
What angle has a sin of ? _______
What quadrant would it be positive and within the range of arcsin? ____
Therefore the angle would be ______
c. sin-1(2) = _________ Sin domain is [-1, 1], therefore No solution
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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, π] 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 trig
function. Ex: sin-1 θ = arcsin θ
Inverse trig functions are NOT equal to the reciprocal of the trig function.
Ex: sin-1 θ ≠ 1/sin θ
There are NO calculator keys for: sec-1 x, csc-1 x, or cot-1 x
And csc-1 x ≠ 1/csc x sec-1 x ≠ 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 unless degrees are asked for.
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Guided practice Example of [E] 28 & 30
Use an inverse trig function
to write θ as a function of x.
28. Cos θ = 4/x so
θ = cos-1(4/x) where x > 0
30. tan θ = (x – 1)/(x2 – 1)
θ = tan-1(x – 1)/(x2 – 1)
where x – 1 > 0 , x > 1
“θ as a function of x” means to write an equation of the form θ equal to an expression 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 θ = 2 so…
draw the triangle
Label the adjacent and opposite sides
Find the hypo. using Pyth. Theorem
So the
θ
2
1
√5
2 5cos
5
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Example
Write an algebraic expression that is equivalent to the given expression.
cos(arctan(1/x))
u
x
1
2
22
1cos
11
x x xu
xx
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
2 1x
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
3tan arccos
5
arccos tan 2
3
2
3 5 5
2
2
1
1
v
v
Word problem involving sin or cos function: P type 1
pcalc643
ALEKS
An object moves in simple harmonic motion with amplitude 12 cm and period 0.1 seconds. At time t = 0 seconds , its displacement d from rest is 12 in a negative direction, and initially it moves in a negative direction.
Give the equation modeling the displacement d as a function of time t.
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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 following function. h(t) = 13 + 6.5 sin 0.25t
In this equation, h(t) is the depth of the water in feet, and t is the time in hours.
Find the following. If necessary, round to the nearest hundredth.
Frequency of h: cycles per hourPeriod of h: hoursMinimum depth of the water: feet UndoUndo HelpClearClear
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