# Trig Review And Error ```Starter
A group of students are asked to express this number
( 00.003048 ) to three significant figures. Which of the
following are correct?
a.
.003
b. .003050
c. 3.048 x 103
d. 3.05 x 10-3
e. .00305
d. and e. are
both correct.
Conversions
Converting From One System of Units to Another
You will need a conversion factor like ( 1 meter = 3.28 ft).
It can be used two ways:
(1m/3.28ft)
or
( 3.28ft/1m)
Multiply your given dimension by the conversion factor to obtain the
desired dimension.
How many feet in 2 meters?
2m (3.28ft/m) = 6.56 feet
How many meters in 10 feet?
10ft(1m/3.28ft) = 3.05 meters
Converting Areas
To convert areas, you must square the conversion factor.
Conversion factor: 1 inch = 2.54cm
A page is 8.5 inches by 11 inches. What is the area in square centimeters?
The area in square inches is 95 in2. So……
95 in2 = __________cm2
95 in2(2.54cm/1 in)2 = 95(6.45 cm2) / (1 in2) = 613 cm2
Converting Volumes
To convert volumes, you must cube the conversion factor.
A cubic foot is how many cubic inches?
Conversion factor: 1 foot = 12 inches
1 ft 3 ( 12 in/ 1 ft)3 = 1 ft 3 ( 123 in3/ 13 ft3) = 1728in3
Using S.I. Prefixes
Examples
Change 12nm to meters.
n = x 10-9 so replace it:
12nm = 12 x 10-9 m
Finished.
Examples
Change 250 grams to kilograms.
1 kg = 1x103 gram
250g ( 1 kg/1x103 g) =
.250 kg
Scientific Notation
If numbers are very large, like the mass of the Earth
5900000000000000000000000 kg
Or very small like the mass of an electron :
.000000000000000000000000000000911 kg
then standard decimal notation is very cumbersome,
so we use scientific notation.
Scientific Notation
A number in scientific notation has two parts:
1st part: a number between 1 and 10
2nd part: 10 to some power.
Example: 5.9 x 1024
1024 Means move the decimal 24 places to the right.
Example: 6.2 x 10-4
10-4 Means move the decimal 4 places to the left.
Examples – Put the number in
Scientific Notation
a. 345000
Answer: 345000 = 3.45 x 105
b. .00034
Answer: .00034 = 3.4 x 10-4
Multiplication and Division
Rule
Example
xmxn = xm+n
xm/xn = xm-n
(xm)n = xmn
(xy)n = xnyn
(x/y)n = xn/yn
x-n = 1/xn
x2x3 = x2+3 = x5
x6/x2 = x6-2 = x4
(x2)3 = x2&times;3 = x6
(xy)3 = x3y3
(x/y)2 = x2 / y2
x-3 = 1/x3
Examples
Simplify:
(2 x 103)(4 x 106)
= (2)(4) x 103(106) = 8 x 109
Simplify:
(4 x 103)/(2 x 106)
= (4)/(2) x 103/106 = 2 x 10-3
Simplify:
(2 x 103)3
= 23 x (103 )3 = 8 x 109
Significant Figures
How to count the number of significant figures in
a decimal number.
Zeros Between other non-zero
digits are significant.
a. 50.3 has three significant
figures
b. 3.0025 has five significant
figures
Significant Figures
Zeros in front of nonzero digits are not
significant:
0.892 has three significant figures
0.0008 has one significant figure
Significant Figures
Zeros that are at the end of a decimal
number are significant.
57.00 has four significant figures
2.000000 has seven significant figures
At the end of a non-decimal number they are not.
5700 has two significant figures
2020 has three significant figures
Non-Decimal Numbers
Major pain to try to figure out the
significant figures – it depends on the
number’s history.
Don’t Use Them.
Practice
Find the number of significant figures.
1.
2.00450
2. .0034050
3.
4.
1450
0.02040
1.
6 sf’s.
2.
5 sf’s
3.
3 sf’s
4.
4 sf’s
Significant Figures
After Division and Multiplication
After performing the calculation,
note the factor that has the least
number of sig figs. Round the product
or quotient to this number of digits.
3.22 X 2.1 = 6.762  6.8
36.5/3.414 = 10.691  10.7
Significant Figures
• Addition or subtraction with significant
figures:
– The final answer should have the same
number of digits to the right of the
decimal as the measurement with the
smallest number of digits to the right of
the decimal.
Ex:
97.3 + 5.85 = 103.15  103.2
Trig Review
20
Sine, Cosine, and Tangent
H
SO
q
sin q = SO / H
SA
cos q = SA / H
tanq = SO / SA
(SO)2 + (SA)2 =H2
21
Example: Find the length of side a and the angles q and f.
f
5
3
q
q
a
a2 + 32 = 52 so a2 = 25 – 9 = 16, or a = 4
4/5 = cosq, so q = cos-1(4/5) = 36.9 degrees
f + q = 90-, so f = 90 – 36.9 = 53.1 degrees
22
Practice
Express each of these in terms of a, b and c.
1. sin(f) = ____________
2. cos(q) = __________
3. sin(q) = _____________
4. tan(f) = ___________
23
Accuracy vs. Precision
Accuracy vs. Precision
• Precision is a measure of how close the
measurements are to each other.
• Poor precision is caused by random errors.
Make many measurements to improve
precision.
• Accuracy is how close the average of the
measurements are to the accepted value.
• Poor accuracy is caused by uncalibrated
measuring devices
Measuring Uncertainties
Two different experimental results, E1
and E2, are measured by percent
difference.
An experimental result E is compared
to an accepted value A, by percent
error.
EXIT
How could you figure out the how tall the
flagpole is that cast this shadow?
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