Homework 17
Problem 1a.
sin(3) = sin(2 + ) = sin 2 cos + cos 2 sin
Problem 1b.
sin(3) = sin(2 + ) = (2 sin cos ) cos + [(cos )2 (sin )2 ](sin ) =
= 3 sin (cos )2 (sin )3
Problem 1c.
(sin )2 + (cos )2
Homework 6
Problem 1.
Problem 2.
4x 3y + 12 = 0 y = 0, 4x = 12, x = 3 x = 0, 3y = 12, y = 4
1
Problem 3.
By examining the graph in problem 2, you can eyeball points that are 5 apart,
for example:
(3,
Homework 1
Problem 1a.
am
1
am
= n = amn
an
a
Problem 1b.
5 (x4 y 3 z 2 )2 (3 z 1 x5 y 3 )1
=
5 (x4 y 3 z 2 )2
5 x8 y 6 z 4
5
5
=
= (x85 y 6(3) z 4(1) ) = x3 y 9 z 5
3 z 1 x5 y 3
3 x5 y 3 z 1
3
3
Pro
Homework 5
Problem 1a.
We know that
8x6 = (2x2 )3
and
27
3
= ( )3
8
2
We apply the formula for dierence of two cubes:
27
3
9
= (2x2 )(4x4 + 3x2 + )
8
2
4
We can still factor the rst factor in the abov
Homework 8
Problem 1.
One should observe that (0.25,0.5) lies on the given line 2x + y = 1. Thus,
the reection is itself, i.e. (0.25,0.5).
Problem 2. We should start by drawing the line L through our
Math W1003, Fall 2013
Lecture 21
Last time, we talked about exponential functions, and dened what we
mean by expressions like:
exp2 (x) = 2x
But quite often, we may be faced with equations like: what
Math W1003, Fall 2013
Lecture 23
Our last topic will be sequences, which will probably also be the rst
topic youll study in calculus. A sequence is a bunch of numbers placed in
a certain order:
2, , e
Homework 14
Problem 1a.
If we shift the graph of sin a by units to the left, we get back the graph of
2
cos a:
sin a +
= cos a
2
If instead we shift the graph of cos a by units to the right, we get si
Homework 18
Problem 1a.
arcsin(1) =
2
2
= 1
5
6
=
4
= 1
sin
because
Problem 1b.
arccos
3
2
=
5
6
because
cos
3
2
Problem 1c.
4
because
tan
2
3
because
cot
arctan(1) =
Problem 1d.
1
arccot
3
=
2
Math 4 Unit 4: Trig Identities
_
Wksht. 4.02-Verifying Identities
Verify the identity algebraically.
1. cscqtan q = secq
3.
csc cos
+
= 2cot
sec sin
5. sin t csct = 1
7.
9.
11.
csc2 x
= csc x sec x
co
Homework 11
Problem 1.
1) at 11:24 pm - thats about the time when the population stopped increasing
2) around 20 million - thats by eyeballing the yvalue when x is about 11:24 pm
3) at 3:36 am - thats
Homework 19
Problem 1.
This is an even function because it is symmetric across the yaxis: f (x) =
f (x).
This is an odd function because it is symmetric across the origin: f (x) =
f (x).
Problem 2.
We
Homework 12
Problem 1
= 30 :
= 45 :
= 60 :
x = x cos(30 ) y sin(30 ) = x 3 y
2 2
y = x sin(30 ) + y cos(30 ) = x + y 3
2
2
x = x cos(45 ) y sin(45 ) = x 2 y 2
2
2
y = x sin(45 ) + y cos(45 ) = x 2
Homework 15
Problem 1a.
5
)
4
x = r cos , y = r sin
(2,
hence:
x = 2 cos
5
4
=2
y = 2 sin
5
4
=2
2
2
2
2
= 2
= 2
So the point has Cartesian coordinates ( 2, 2)
1
Problem 1b.
5
)
3
x = r cos , y = r
Math W1003, Fall 2013
Practice Midterm Exam
2 hours, 4 problems, 100 points. No calculators or notes/books, but
you are welcome (though not required) to use graph paper/ruler.
Explain all steps in you
Math W1003, Fall 2013
Practice Final 1
3 hours, 6 problems, 150 points. No calculators or notes/books, but
you are welcome (though not required) to use graph paper/ruler.
Explain all steps in your rea
Math W1003, Fall 2013
Lecture 13
We will now start talking about the modern way to think about trigonometric functions, and the rst step in doing so is to stop thinking of angles
as measured in degree
Math W1003, Fall 2013
Lecture 7
In our last class we discussed lines in the coordinate plane. The equation of
a line is:
ax + by + c = 0
(1)
for some real constants a, b, c. This should be read as:
th
Math W1003, Fall 2013
Lecture 8
We will start this class by doing a bonus topic to our previous lecture.
Namely, we gave formulas for the reection in certain special lines (the axes
x = 0 and y = 0, a