Math 140
Lecture 13
x
Recall: Since log b x and b are inverses of each other:
v log b x y iff x b y
v log b b x x
v b logb x x
logb1 = 0, ln(1) = 0
logbb = 1, ln(e) = 1
Write as a sum / difference / multiple of the simplest
possible logarithms.
`logb 4
2y
Math 140
Hw 14
Worked examples of selected recommended problems.
Write N(t) in base e form. Then solve the problem.
A. A colony of bacteria starts with 2000 bugs. Two hours
later it has 3800.
Given: N 0 2000
N2 3800
Nt N 0 e kt
tN2 2000e k2
t2000e 2k 3800
Math 140
Hw 17
Worked examples of selected recommended problems.
sin
2
Acos 2 A
A. sin Acos A
Hint: factor the top.
sin A cos Asin A cos A/sin A cos A
sin A cos A
B. sin 2 cos csc 3 sec
1
1
sin 2 cos sin 3 cos
1
sin
csc
cot A sin A cos A cot A
=
Math 140
Hw 18
Worked examples of selected recommended problems.
(a) Rewrite in terms of an angle in [0,p/2].
(b) Find the exact answer, no decimals.
E. Simplify
A(a). cos11/6
cos11/6 cos12/6 /6
cos2 /6 cos/6 cos/6
1
sec 2 t
sin 2 tcos 2 t
tan 2 t1
cos
Math 140
Hw 15
Worked examples of selected recommended problems.
Remember to include any needed units, e.g., mi/hr, rad/sec. Except
for 5ab, give only exact answers. No decimals.
A. Convert 45o to radian measure.
45 o 180
4
G. Find the six trigonometric f
Math 140
Hw 16
Worked examples of selected recommended problems.
For each angle, sketch the reference angle, draw an
arrow to indicate the angle's direction. A reference
angle must be positive and < /2 = 90o.
A. (a) 110o
(b) 60o
60
70
sin cos
F. Simplify
Math 140
Hw 19
Worked examples of selected recommended problems.
Find the period and amplitude.
A.
period = 4 amplitude= 6
F. y 3 sin (x/2
amplitude= 3
6
6
2
3 4
2
period= 4
3
5
1
4
xintercepts: 0, 2, 4
0
increases on [0, 1], [3, 4]
B.
period = 4
amplit
Math 140
Hw 24
Worked examples of selected recommended problems.
Evaluate exactly without a calculator: p and 2 rather than
3.14 or 1.41. Answer may by undef.
K. cosarcsin 2
7
5 symbol fraction with square root
1 2 2
7
1
A. sin 3 /2
1
Answer: /3
B. tan
Math 140
Hw 21
Worked examples of selected recommended problems.
(1) Simplify using an addition formula (show the step
immediately after the addition formula). First point.
(2) Complete the simplification.
Second point.
Examples:
`sinx cosy x cosx siny x
Math 140
Hw 22
Worked examples of selected recommended problems.
Evaluate the expressions.
A. sin 3 .
4
2
First find cos q. The quadrant determines the sign.
2 quadrant II cos 0
9
7
cos 1 sin 2 1 16 16
(c) tan/12
tan
sin 2 2 sin cos 2 3
4
7
4
7
4
cos 2
Math 140
Hw 20
Worked examples of selected recommended problems.
(1) Graph over one period (period doesn't have to start at 0). (2) List the xintercepts. (3) List both coordinates of the
highest and lowest points. Get the box from the amplitude, period a
Math 140
Hw 23
Worked examples of selected recommended problems.
Write each expression as a sum or difference of
trigonometric functions. Don't calculate the answer.
A. sin 20 cos 10
1 sin20 0 10 0 sin20 0 10 0
2
1
sin10 0 sin30 0
2
1 sin10 0 1
2
2
Math 140
Hw 26
Worked examples of selected recommended problems.
(a) Give the unsimplified exact answer obtained using the
law of sines or cosines or Straight V Thm. E.g.
a 8 2 5 2 285 cos20 0 or b
5 sin100 o
sin50 o .
An unsimplified exact answer for an
Math 140
Hw 28
Worked examples of selected recommended problems.
Graph the parabola. On the graph, mark and give the
coordinates for focus and vertex. Draw the directrix with
a dotted line.
First locate the focus and vertex and draw the directrix. Then ma
Math 140
Hw 27
Worked examples of selected recommended problems.
D. 4, 11/6
x, y r cos , r sin 4 cos , 4 sin
6
= 4 3 /2, 41/2 2 3 , 2
E. 4, /6
x, y r cos , r sin 4 cos , 4 sin
6
=4 3 /2, 41/2 2 3 , 2
C
b
A
a
B
c
A(a). sinB = 1/ 2 ,
What are the possible v
Math 140
Lecture 31 (omitted)
Note the difference between parabolas, ellipses, and
hyperbolas.
v If only one variable is squared, it is a parabola.
Write with the squared variable on the left.
x a 2 ky b Vertical parabola like y = x2 or y =
x2
y b 2 kx a
Math 140
Hw 0
Recommended problems, don't turn this in.
Rewrite each expression in a form that does not contain
absolute values.
A. x 3 x 4 where x 3.
Write the following in interval notation.
I. x 4.
J. x 1.
B. x 3 x 4 where 3 x 4.
K. x 5 3
Math 140 Practice Exam 2 /100
Exam 2. Lectures 712. No calculators. Understanding is not
enough; you must be proficient enough to complete the exam in 50
minutes.
Know the area and volume formulas for triangles, rectangles,
circles, boxes, cans (includin
Math 140 Lecture 17
2
2
RECALL. sin q + cos q = 1
tan q = sin q/cos q
cot q = cos q/sin q
sec q = 1/cos q
csc q = 1/sin q
Simplify.
`cot sec cos
`
sin x 1
sin x 1
sin x 1
sin x 1
cos 1
sin cos
sin x1sin x
sin x1sin x
Proof. In the picture below,
cos cot
Math 140
sin and cos have period 2, but tan has period .
Lecture 18
Exam 3 covers lectures 13 18. Study the recommended exercises.
DEFINITION. For any function f: f is even iff f (x) = f (x).
f is odd iff f (x) =  f (x).
Graphically, the left half (th
Math 140
Lecture 21
RECALL. sin2+ cos2 = 1, sin(x) = sin(x), cos(x) = cos(x).
cos(p/2x) = sin(x), sin(p/2x) = cos(x).
`cos
TRIGONOMETRIC ADDITION FORMULAS.
sins t sin s cos t cos s sin t
coss t cos s cos t sin s sin t
coss t cos s cos t sin s sin t
3
Math 140
`cos 1 , 3 2.
2
3
Evaluate sin(2q), cos(2q), sin(q/2), cos(q/2).
First find sin(q). The inequality quadrant IVsin q<0.
Lecture 22
DOUBLEANGLE FORMULAS.
sin 2 2 sin cos
cos 2 cos 2 sin 2
tan 2
sin 1 cos 2 1
2 tan
1tan 2
Proof.
sin 2 sin
s
Math 140
Lecture 24
Exam 4: Lectures 1924, including this lecture.
Inverse trigonometric functions
RECALL. A function is 11 iff no horizontal line crosses its
graph more than once.
While not 11 in general, sin, cos, and tan are 11 on the
first quadran
Math 140
Since tan sin / cos 1 /
2
6
6
6
Producttosum rules
RECALL
cosx y cos x cos y sin x sin y
cosx y cos x cos y sin x sin y
Adding these gives
cosx y cosx y 2 cos x cos y
Subtracting gives
cosx y cosx y 2 sin x sin y
Similarly
sinx y sin x cos y co
Math 140
Lecture 19
Graphs of sin and cos
RECALL. sin and cos have period 2p:
sin(x +2p) = sin(x), cos(x +2p) = cos(x).
We often graph periodic functions only over one period,
e.g., [0, 2p]. Before and after this interval, they repeat.
Note sin(x) = 0 iff
Math 140
vert. asymptotes: ., x = p/2, x = p/2, x = 3p/2, . .
Lecture 20
RECALL. For C > 0, the graph of
f (x + C) is the graph of f (x) shifted C units to the left.
f (x C) is the graph of f (x) shifted C units to the right.
The phase shift is the amou
Math 140
Lecture 29
v
`Find the focus, directrix and graph of y 2y 4x 5.
y 2 2y 1 4x 4, y 1 2 4x 1
p k/4 4/4 1.
This is the parabola y 2 4x
shifted: down 1 unit, right 1 unit.
The y 2 means the parabola is horizontal.
(1,1).
(2,1).
x=0.
y2
b2
1 is a ho
Math 140
Lecture 26
CONVENTION. Assume side a is opposite angle A, side b is
opposite angle B and side c is opposite angle C.
SINE LAWS. In any triangle, the ratio of one angles sine
and its opposite side equals the ratio of any other
angles sine and oppo
Math 140
To graph, complete the squares if necessary, then write
the equation in one of the above forms.
Lecture 30
horizontal hyperbola
ve
rte
x

y2
x
cal a
fo
is
x2
4
1,
y2
12
x2
22
1
a 1, b 2, c 1 4 5
Vertical hyperbola (the y 2 is positive).
Majo