Objectives: Assignment: To define, graph, and P. 349: 1-16 S

Objectives:
1.
To define, graph, and use the inverses of sine, cosine, and tangent
Assignment:
•
P. 349: 1-16 S
•
P. 349: 19-34 S
•
P. 349: 37-42 S
•
P. 349: 43-48 S
•
P. 350: 49-58 S
•
P. 350: 59-64 S
•
P. 350-2: 71, 96, 105
You will be able to define, graph, and use the inverse of sine, cosine, and tangent
What does an inverse do to a function algebraically and graphically?
Inverses switch inputs and outputs Inverses reflect a graph over y = x
Inverses give you a way to find the input when you know the output.
Explain what x and y represent in y = sin x, then explain what is meant by the inverse of
y
= sin x.
Use the graph of y = sin x to explain why its inverse is not a function.
The inverse of a function f is also a function iff no horizontal line intersects the graph of f more than once.
What could we do to the graph below so that its inverse is cleverly a function?
We can restrict the domain of the function to an interval that would pass the horizontal line test.
In order for the inverse of y = sin x to be a function, we have restrict the domain. Refer again to the graph of f (x) = sin x and find a sensible interval for which f
−1
(x) is a function.
While any number of intervals would work, the most convenient is [−π/2, π/2].
Increasing, has a full range of values, contains all the acute angles of a triangle, 1-1
The inverse sine function is defined by
y
y
x x
1 and
2
y
2
The inverse sine function is defined by
y
arcsin
x
Read “the arcsine of x”
Means “the arc whose sine is x”
The inverse sine function is defined by
y
1 sin
x
if and only if sin
x
1 and
2
y y
x
2
The inverse sine function is defined by
y
sin
1
x
Read “the inverse sine of x”
Means “the angle whose sine is x”
Does not mean reciprocal
The inverse sine function is defined by
y
sin
1
x
Domain:
1,1
Sine ratio
y
arcsin
x
Range:
,
2 2
Angle measure
Find the exact value of each of the following:
1.
arcsin
1
Note that there are infinitely many values of
2.
sin
1
2
3
x
for which sin x = ½ , but there is only one in the interval [−π/2, π/2].
3.
4.
sin
1
Graph y = arcsin x over its entire domain!
To do this, find the values of y = sin x and exchange x and
y
.
Refer to the graph of f (x) = cos x and find a sensible interval for which f
−1
(x) is a function.
The inverse cosine function is defined by
y
y
x x
1 and 0
Refer to the graph of f (x) = tan x and find a sensible interval for which f
−1
(x) is a function.
The inverse tangent function is defined by
y
where
x
and
2
y
x y
2
Find the exact value of each of the following:
1.
arccos
1
2.
cos
1
2
3
3.
arctan
4.
tan
1
Use a calculator to approximate each of the following.
1.
arctan (4.84)
2.
arccos (−.349)
3.
sin
−1
(1.1)
4.
arcsin (.321)
If a relation and its inverse are both functions, then they are called
inverse functions
.
( )
x
and
( )
x
However, the composition of a trig function and its inverse does not always give you x!
You have to make sure the domains and ranges match up properly.
x x
1
y
2
y
2
x x
1
x
x
y
0
y
y
2
y
2
If possible, find the exact value.
1.
14
2.
3.
4.
arcsin sin
5
3
To make these compositions a bit more fun, what if we mismatched the trig functions with their inverses?
To do these problems, draw a right triangle and use the Pythagorean Theorem.
csc arctan
5
12
13
5 tan
5
12
Quadrant IV tan
5
12
Quadrant II
Find the exact value of each of the following.
1.
cos arctan
3
4
2.
sin arccos
2
When you take Calculus and you finally learn to integrate, you’ll sometimes have to turn an inverse trig expression into an algebraic one.
cos
x
3
x
2
Quadrant I
2
Write each of the following as an algebraic expression in x.
1.
2.
Objectives:
1.
To define, graph, and use the inverses of sine, cosine, and tangent
Assignment:
•
P. 349: 1-16 S
•
P. 349: 19-34 S
•
P. 349: 37-42 S
•
P. 349: 43-48 S
•
P. 350: 49-58 S
•
P. 350: 59-64 S
•
P. 350-2: 71, 96, 105
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