Transcript Unit01 PowerPoint for trigonometry class

Welcome
to Week 1
College
Trigonometry
Trigonometry?
What is trigonometry?
Trigonometry?
Tri means three
Gon means sides
Ometry means measurement
Trigonometry?
Originated with the Egyptians
Trigonometry?
Arabic Trigonometry was
developed in order to
observe holy days on
the correct days in
all parts of the
Islamic world
Trigonometry?
There was also a need for nonnavigators to be able
to travel to Mecca
each year and return
successfully
Trigonometry?
In the early 9th century AD,
Muhammad ibn Mūsā
al-Khwārizmī wrote
the first modern
trig book
(blame him…)
Trigonometry?
The trig functions and natural
exponentials and logarithms we
study in this class are needed
in most formulas used to
describe how our complicated
universe works
Trigonometry?
astronomy
banking
electronics
biology
atoms
forensics
construction
...
Trigonometry!
At last!
Something USEFUL!!!
Questions?
Angles
To study triangles, let’s start
with one corner:
an angle
Angles
A ray is half of a line
it has an origin
the other end stretches
on forever
origin
Sun rays
Death rays
Angles
Real-life death rays:
Gamma rays
Cosmic rays
Angles
If two rays start at the same
origin, they form an “angle”
Angles
Their point of common origin is
called the “vertex”
vertex
Angles
Angles are usually represented
by lowercase Greek letters:
α, β, θ
θ
Angles
Angles have an
initial (beginning) side and a
terminal (ending) side
terminal
θ
initial
Angles
The “standard position” for an
angle: vertex at origin and
initial side along x-axis
θ
(0,0)
x
Angles
Positive angles counterclockwise rotation from
initial side
θ
Angles
Negative angles - clockwise
rotation from initial side
θ
ANGLES
IN-CLASS PROBLEMS
Which of the following graphs
represent negative angles?
Angles
Remember graph paper?
Angles
Graph paper is made of angles:
Angles
Graph paper is
split into 4
quadrants:
Angles
The quadrants move
counterclockwise
around the grid
II
just like positive
angles
III
I
IV
Angles
An angle with a terminal side in
a quadrant
"lies" in that
I
II
quadrant
III
IV
ANGLES
IN-CLASS PROBLEMS
Which quadrant does each angle
lie?
Angles
An angle with a terminal side
along any axis is called
“quadrantal”
Angles
Angles are measure by
determining the amount of
rotation from the initial side to
the terminal side
θ
Angles
A degree, symbolized by °
measures angles
θ
23°
ANGLES
IN-CLASS PROBLEMS
Which of the graphs represents
the angle θ = –45º in standard
position?
Angles
Types of angles:
ANGLES
IN-CLASS PROBLEMS
Classify the angle:
ANGLES
IN-CLASS PROBLEMS
Classify the angle:
168º
42º
90º
180º
263º
Angles
A “reference angle” is a
positive acute angle θ° formed
by the terminal side of a
nonacute angle θ and the x-axis
ANGLES
IN-CLASS PROBLEMS
Which is the reference angle?
ANGLES
IN-CLASS PROBLEMS
Co-terminal angle θ ± k•360°
Which angle is co-terminal to
275º?
a) 85º
c) –55º
b) –85º
d) 55º
ANGLES
IN-CLASS PROBLEMS
Complementary - two positive
angles whose sum = 90°
Find the complementary angle
to 32º
a) 148º
c) 328º
b) –32º
d) 58º
ANGLES
IN-CLASS PROBLEMS
Supplementary - two positive
angles whose sum = 180°
Find the supplementary angle to
32º:
a) 148º
c) 328º
b) –32º
d) 58º
ANGLES
IN-CLASS PROBLEMS
Explementary - two positive
angles whose sum = 360°
Find the explementary angle to
32º:
a) 148º
c) 328º
b) –32º
d) 58º
Questions?
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
A polygon with three sides and
three angles is called _______
Right Angle Triangles
A triangle!
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
A special triangle has one
corner that is a right angle –
what is it called?
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
The side of the triangle
opposite of the right angle is
called:
__________
Right Angle Triangles
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
Given the lengths of two sides
of a right triangle, you can
calculate the third using ____
a2 + b2 = c2
Right Angle Triangles
Pythagorean
Theorem video
RIGHT ANGLE TRIANGLES
IN-CLASS PROBLEMS
Find the missing side:
a = 4
b = 3
c = _____
a = _____
b = 3
c = 5
a = 4
b = _____
c = 5
a = 6
b = 2
c = _____
a = 1
b = 2
c = _____
a = 2
b = 1
c = _____
Questions?
Trigonometry
Based on a right triangle:
Trigonometry
Trigonometry came to us from
the ancient Greeks who drew
triangles in circles
Trigonometry
There are six trig functions:
sine = length of side opposite θ
length of hypotenuse
Trigonometry
cosine = length of side adjacent θ
length of hypotenuse
Trigonometry
tangent = length of side opposite θ
length of side adjacent θ
Trigonometry
cotangent = length of side adjacent θ
length of side opposite θ
secant
= length
of
hypotenuse
length of side adjacent θ
cosecant
= length
of
hypotenuse
length of side opposite θ
Trigonometry
US vs
Europe
bakerfamilytree.blogspot.com
Trigonometry
Trigonometry
Trigonometry
Based on the formula a2+b2=c2,
where c is the hypotenuse and
a and b are the other sides,
there are three identities:
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
Trigonometry
Hipparchus calculated the first
trig table
Trigonometry
Cot/Sec/Csc Stickers
TRIGONOMETRY
IN-CLASS PROBLEMS
Find these trig functions using
your calculator:
sin 10º
sin 90º
cos 45º
tan 0º
sin 0º
cos 20º
cos 90º
tan 45º
sin 45º
cos 0º
tan 30º
tan 90º
TRIGONOMETRY
IN-CLASS PROBLEMS
Evaluate tan 30º
a)
𝟑
c) –1
b)
d) 1
𝟐
TRIGONOMETRY
IN-CLASS PROBLEMS
Evaluate sec 45º
a)
𝟑
c) –1
b)
d) 1
𝟐
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 3/5 and
cos θ = 4/5, find tan θ
a) 4/5
c) 5/3
b) 3/4
d) 5/4
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 2/3 and
cos θ =
5/3, find cot θ
a) 1
b)
5/2
c) 2/ 5
d) 3/2
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 3/5 and
cos θ = 4/5, find csc θ
a) 4/5
c) 5/3
b) 3/4
d) 5/4
TRIGONOMETRY
IN-CLASS PROBLEMS
Given sin θ = 2/3 and
cos θ = 5/3, find sec θ
a) 1
c) 5/3
b) 3/5
d) 5/2
TRIGONOMETRY
IN-CLASS PROBLEMS
Which of the following
expressions represents the
same value as csc 35º?
a) sec 55º
c) tan 55º
b) cos 55º
d) cot 55º
TRIGONOMETRY
IN-CLASS PROBLEMS
Simplify sin2 10º + cos2 10º
a) 2
c) 3
b) 1
d) 10
Trigonometry
TRIGONOMETRY
HOMEWORK PROBLEM
The wife of the victim said
that she had just asked him for
a divorce when he suddenly
pulled a .357 out of his jacket
pocket and shot himself in the
head.
TRIGONOMETRY
HOMEWORK PROBLEM
According to her statement, he
was standing beside the kitchen
sink at the time of the shot.
The victim has a single, nearcontact entrance wound above
his right ear 67” above the
heel. The shot did not exit.
TRIGONOMETRY
HOMEWORK PROBLEM
Determine the height of impact
to determine if the wife is
telling the truth or not.
TRIGONOMETRY
HOMEWORK PROBLEM
Blood spatter pattern:
A B C and D are the angles of
impact, the other numbers are
the distance to the point of
convergence
TRIGONOMETRY
HOMEWORK PROBLEM
Calculate the height of impact
using the following equation:
tangent of the
distance from
angle of impact * base of spatter to
θ
pt. of convergence
height of
=
impact
TRIGONOMETRY
HOMEWORK PROBLEM
Angle
STAIN
of
Impact θ
tan(θ)
Distance (D)
to Point of
Convergence
A
62º
20.5"
B
42º
34.5"
C
29º
51.375"
D
22º
76.25"
(D) *
tan(θ)
TRIGONOMETRY
HOMEWORK PROBLEM
1) Sum of (D)*tan(θ) = _____
2) Number of Stains = _____
3) Height of impact = 1)/2)
= _____
TRIGONOMETRY
HOMEWORK PROBLEM
If the victim's ear was 67”
above the floor, was the wife
telling the truth?
Questions?
Liberation!
Be sure to turn in your
exercises to me before you
leave
Don’t forget
your lab homework
due next week!
Have a great
rest of the week!