Upload
ellis
View
88
Download
4
Embed Size (px)
DESCRIPTION
Trig/Precalc Chapter 4.7 Inverse trig functions. Objectives Evaluate and graph the inverse sine function Evaluate and graph the remaining five inverse trig functions Evaluate and graph the composition of trig functions. y = sin(x). - π /2. π /2. π. 2 π. - PowerPoint PPT Presentation
Citation preview
1
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
2
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
3
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
5
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 ______
6
IV6
6
6
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
3
32
32
32
√3 2
1
I
3
3
No Solution
7
Graphs of Inverse Trigonometric Functions
The basic idea of the arc function is the same whether it is arcsin, arccos, or arctan
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
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 ? _______
10
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
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 unless degrees are asked for.
12
Guided practice Example of [E] 28 & 30Use an inverse trig functionto 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
13
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 5cos5
14
Example
Write an algebraic expression that is equivalent to the given expression.
cos(arctan(1/x))
u
x
1
2
22
1cos11
x x xuxx
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:
3arcsin2
3arcsin sin2
3tan arccos5
arccos tan 2
3
2
3 5 5
2
2
11
vv
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.
Undo HelpClear
Next >> Explain
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 Undo HelpClear
Next >> Explain