Transcript 3-8 IOT

UNIT 3 – ENERGY AND POWER
Topics Covered
HAVE– Fuels
PAPER
1. Energy Sources
and Power Plants
PROTRACTORS/RULERS
FOR
2.
Trigonometry and Vectors
3. Classical Mechanics:
STUDENTS
WITHOUT THEM, SELL
Force, Work, Energy, and Power
THEM
FOR
$0.25
/
$0.50
4. Impacts of Current Generation and Use
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Background – Trigonometry
1.
2.
3.
4.
5.
6.
7.
8.
Trigonometry, triangle measure, from Greek.
Mathematics that deals with the sides and angles of triangles,
and their relationships.
Computational Geometry (Geometry – earth measure).
Deals mostly with right triangles.
Historically developed for astronomy and geography.
Not the work of any one person or nation – spans 1000s yrs.
REQUIRED for the study of Calculus.
Currently used mainly in physics, engineering, and chemistry,
with applications in natural and social sciences.
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometry
1.
2.
3.
4.
Total degrees in a triangle: 180
Three angles of the triangle below: A, B, and
Three sides of the triangle below: a, b, and c
Pythagorean Theorem:
B
C
a2 + b2 = c2
c
A
b
a
C
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometry
State the Pythagorean Theorem in words:
“The sum of the squares of the two sides of a right triangle is
equal to the square of the hypotenuse.”
Pythagorean Theorem:
B
a2 + b2 = c2
c
A
b
a
C
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometry – Pyth. Thm. Problems
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
1. Solve for the unknown hypotenuse of the following triangles:
a)
b)
?
3
4
?
c)
1
?
1
3
1
c2  a 2  b2
2
2
2
2
c a b
c a b
c  a 2  b2 Align equal
signs
possible
2
2
2 when
2
 ( 3)  1
 1 1
 9  16
 3 1
c

2
c5
c2
Trigonometry and Vectors
Common triangles in Geometry and
Trigonometry
5
1
3
4
Trigonometry and Vectors
Common triangles in Geometry and
Trigonometry
You must memorize these triangles
45o
60o
2
2
1
30o
45o
1
2
1
3
3
Trigonometry and Vectors
Trigonometry – Pyth. Thm. Problems
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
2. Solve for the unknown side of the following triangles:
a)
10
b)
8
?
c a b
Divide
all2 sides by 2
2
2
a  c3-4-5
 b triangle
a  c2  b2
2
2
2
 102  82
 36
13
a 6
?
c)
12
?
12
15
a  c2  b2
a  c2  b2
 132 122
 169  144
 25
a 5
Divide
all2sides
 15
 122 by 3
3-4-5
triangle
 225
 144
 81
a 9
Trigonometry and Vectors
Trigonometric Functions – Sine
Standard triangle labeling.
Sine of <A is equal to the side opposite <A divided by the
hypotenuse.
opposite
sin A =
hypotenuse
a
sin A =
c
B
c
A
ADJACENT
b
OPPOSITE
1.
2.
a
C
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometric Functions – Cosine
Standard triangle labeling.
Cosine of <A is equal to the side adjacent <A divided by the
hypotenuse.
adjacent
cos A =
hypotenuse
cos A =
b
c
B
c
A
ADJACENT
b
OPPOSITE
1.
2.
a
C
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3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometric Functions – Tangent
Standard triangle labeling.
Tangent of <A is equal to the side opposite <A divided by the
side adjacent <A.
tan A =
opposite
adjacent
tan A =
a
b
B
c
A
ADJACENT
b
OPPOSITE
1.
2.
a
C
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3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometric Function Problems
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
3. For <A below calculate Sine, Cosine, and Tangent:
B
c)
b)
a)
B
5
3
4
C
2
B
2
1
Sketch and answer in your notebook
A
opp.
sin A =
hyp.
A
1
C
tan A =
opp.
adj.
A
3
cos A =
1
C
adj.
hyp.
Trigonometry and Vectors
Trigonometric Function Problems
3. For <A below, calculate Sine, Cosine, and Tangent:
a)
A
B
5
4
opposite
sin A =
hypotenuse
3
C
3
sin A =
5
adjacent
cos A =
hypotenuse
4
cos A =
5
opposite
tan A =
adjacent
3
tan A =
4
Trigonometry and Vectors
Trigonometric Function Problems
3. For <A below, calculate Sine, Cosine, and Tangent:
B
b)
A
2
1
1
C
opposite
sin A =
hypotenuse
1
sin A =
√2
adjacent
cos A =
hypotenuse
cos A = 1
√2
opposite
tan A =
adjacent
tan A = 1
Trigonometry and Vectors
Trigonometric Function Problems
3. For <A below, calculate Sine, Cosine, and Tangent:
B
c)
A
2
1
3
C
opposite
sin A =
hypotenuse
1
sin A =
2
opposite
tan A =
adjacent
adjacent
cos A =
hypotenuse
tan A = 1
√3
cos A = √3
2
Trigonometry and Vectors
Trigonometric Functions
Trigonometric functions are ratios of the lengths of the
segments that make up angles.
opposite
sin A =
hypotenuse
adjacent
cos A =
hypotenuse
tan A =
opposite
adjacent
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Common triangles in Trigonometry
You must memorize these triangles
45o
60o
2
2
1
1
30o
45o
1
3
Trigonometry and Vectors
Trigonometric Functions
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
4. Calculate sine, cosine, and tangent for the following angles:
a. 30o
1
b. 60o
sin 30 =
o
2
60
c. 45o
2
cos 30 = √3
2
tan 30 = 1
√3
1
30o
3
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometric Functions
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
4. Calculate sine, cosine, and tangent for the following angles:
a. 30o
√3
b. 60o
sin 60 =
o
2
60
c. 45o
2
1
cos 60 =
2
tan 60 = √3
1
30o
3
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Trigonometric Functions
NO CALCULATORS – SKETCH
– SIMPLIFY ANSWERS
4. Calculate sine, cosine, and tangent for the following angles:
a. 30o
b. 60o
45o
1
cos 45 =
o
√2
c. 45
2
1
sin 45 =
√2
tan 45 = 1
1
45o
1
IOT
3-8
POLY ENGINEERING
Trigonometry and Vectors
Measuring Angles
Unless otherwise specified:
• Positive angles measured counter-clockwise from the horizontal.
• Negative angles measured clockwise from the horizontal.
• We call the horizontal line 0o, or the initial side
90
30 degrees = -330 degrees
45 degrees = -315 degrees
90 degrees = -270 degrees
180
INITIAL SIDE
0
180 degrees = -180 degrees
270 degrees = -90 degrees
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270
360 degrees
3-8
POLY ENGINEERING
Trigonometry and Vectors
•
•
•
•
Begin all lines as light construction lines!
Draw the initial side – horizontal line.
From each vertex, precisely measure the angle with a protractor.
Measure 1” along the hypotenuse. Using protractor, draw vertical
line from the 1” point.
Darken the triangle.
Trigonometry and Vectors
CLASSWORK / HOMEWORK
Complete problems 1-3 on the
Trigonometry Worksheet