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Standards Covered
•
MCC9‐12.N.VM.10 (+) Understand that the zero and identity matrices play a role in
matrix addition and multiplication similar to the role of 0 and 1 in the real numbers.
The determinant of a square matrix is nonzero if and only if the matrix has a
multiplicative inverse.
• MCC9‐12.A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems
of linear equations (using technology for matrices of dimension 3 × 3 or greater).
• MCC9‐12.A.REI.8(+)Represent a system of linear equations as a single matrix equation
in a vector variable.
• MCC9 ‐ 12.A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems
of linear equations (using technology for matrices of dimension 3 × 3 or greater).
AccPreCalc Lesson 6 Angles/ The Unit
Circle (aM3 43)
Essential Question: How can I use circles and degrees
to understand the unit circle? What makes an angle
positive or negative? What are conterminal angles?
How can angle measures be greater than 180
degrees?
Standards: Extend the domain of trigonometric
functions using the unit circle
MCC9 ‐ 12.F.TF.3 (+) Use special triangles to determine
geometrically the values of sine, cosine, tangent for π
/3, π /4 and π /6, and use the unit circle to express
the values of sine, cosine, and tangent for π ‐ x, π +x,
and 2 π ‐ x in terms of their values for x, where x is
any real number.
New Vocabulary
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Acute Angles- between 0 and 9 degrees
Right angles- 90 degrees
Obtuse angles- between 90 and 180 degrees
Straight angles- are equal to 180 degrees
Complementary angles- 2 angles whose sum is 90
degrees
• Supplementary angles- 2 angles whose sum is
180 degrees
• coterminal angles- angles that have the same
terminal ray
Important formulas
• D = diameter
• r = radius
D = 2r or 1.2 D = R
Eratosthenese
• He is best known for being the first person to
calculate the circumference of the earth.
Pi ≈ 3.142
Note: pi is irrational
Pi = c / d
this means π cannot be expressed exactly as a ratio of any two
integers (fractions such as 22/7 and other rational numbers are
commonly used to approximate π, but no ratio of integers can be
its exact value).
More Formulas
C = 2 pi r
Where
C = Circumference
And r = radius
Proof:
We know
pi = C / d
Since d = 2r
Pi = C/ 2r
Multiply both sides
by 2r to get:
2 pi r = C
Q.E.D
Area of a Circle
Example:
A = ∏ r2
Where A = area of a
circle
And r = radius
r=7
A = ∏ r2
A = ∏ (49)
Formulas on one page
d = 2r
or
r = (1/2) d
c = 2 pi r
pi = c / d
A = pi
2
r
Practice Problem
Given:
Area of a circle = 49 pi in2
Find: circumference
we know A = pi r2
so,
49 pi = pi r2
divide both sides
by pi to get
49 = r2
take the square root of both sides to get
r = 7 in
We also know
C = 2 pi r
since r = 7
C = 2 (7) pi
C = 14 pi
inches
Θ ≈ 53 degrees
Θ is a central angle
Arc
If you go counter clockwise you get a
Positive angle.
If you go clock wise you get a negative angle.
Coterminal angles
• coterminal angles- angles that have the same
terminal ray
270 degrees and -90 degrees
are coterminal angles
Quadrants
Let's say we have an angle β = 600
degrees
360 degrees is once
around
then we have 270
degrees left to go
This puts us in Q3
coterminal angles
600 degrees
240 degrees
-120 degrees
960 degrees
You add or subtract 360 to
get a coterminal angle
Arc measure = central angle measure