TRIGONOMETRY

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TRIGONOMETRY

• http//math.la.asu.edu/tdalesan/mat170/TRIGONOMET
  RY.ppt

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Angles, Arc length, Conversions
Angle measured in standard position. Initial
side is the positive x axis which is fixed.
Terminal side is the ray in quadrant II, which
is free to rotate about the origin.
Counterclockwise rotation is positive, clockwise
rotation is negative.
Coterminal Angles Angles that have the same
terminal side. 60, 420, and 300 are all
coterminal.
Degrees to radians Multiply angle by
radians
Radians to degrees Multiply angle by
Note 1 revolution 360 2p radians.
Arc length central angle x radius, or
Note The central angle must be in radian measure.
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Right Triangle Trig Definitions
B
c
a
A
C
b

• sin(A) sine of A opposite / hypotenuse a/c
• cos(A) cosine of A adjacent / hypotenuse
  b/c
• tan(A) tangent of A opposite / adjacent a/b
• csc(A) cosecant of A hypotenuse / opposite
  c/a
• sec(A) secant of A hypotenuse / adjacent
  c/b
• cot(A) cotangent of A adjacent / opposite
  b/a

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Special Right Triangles
30
45
2
1
60
45
1
1
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Basic Trigonometric Identities


Quotient identities
Even/Odd identities
Even functions
Odd functions
Odd functions

Reciprocal Identities

Pythagorean Identities
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All Students Take Calculus.
Quad I

• Quad II

cos(A)gt0 sin(A)gt0 tan(A)gt0 sec(A)gt0 csc(A)gt0 cot(A
)gt0
cos(A)lt0 sin(A)gt0 tan(A)lt0 sec(A)lt0 csc(A)gt0 cot(A
)lt0
cos(A)lt0 sin(A)lt0 tan(A)gt0 sec(A)lt0 csc(A)lt0 cot(A
)gt0
cos(A)gt0 sin(A)lt0 tan(A)lt0 sec(A)gt0 csc(A)lt0 cot(A
)lt0
Quad IV
Quad III
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Reference Angles
Quad I
Quad II
? ?
? 180 ?
? p ?
? ? 180
? 360 ?
? 2p ?
? ? p
Quad III
Quad IV
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Unit circle

• Radius of the circle is 1.
• x cos(?)
• y sin(?)
• Pythagorean Theorem
• This gives the identity
• Zeros of sin(?) are where n is an integer.
• Zeros of cos(?) are where n is an integer.

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Graphs of sine cosine

• Fundamental period of sine and cosine is 2p.
• Domain of sine and cosine is
• Range of sine and cosine is AD, AD.
• The amplitude of a sine and cosine graph is A.
• The vertical shift or average value of sine and
  cosine graph is D.
• The period of sine and cosine graph is
• The phase shift or horizontal shift is

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Sine graphs
y sin(x)
y sin(x) 3
y 3sin(x)
y sin(3x)
y sin(x 3)
y 3sin(3x-9)3 y sin(x)
y sin(x/3)
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Graphs of cosine
y cos(x)
y cos(x) 3
y 3cos(x)
y cos(3x)
y cos(x 3)
y 3cos(3x 9) 3 y cos(x)
y cos(x/3)
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Tangent and cotangent graphs

• Fundamental period of tangent and cotangent is p.
  
• Domain of tangent is where n is an
  integer.
• Domain of cotangent where n is an
  integer.
• Range of tangent and cotangent is
• The period of tangent or cotangent graph is

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Graphs of tangent and cotangent
y tan(x) Vertical asymptotes at
y cot(x) Verrical asymptotes at
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Graphs of secant and cosecant
y csc(x) Vertical asymptotes at Range (8, 1
U 1, 8) y sin(x)
y sec(x) Vertical asymptotes at Range (8, 1
U 1, 8) y cos(x)
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Inverse Trigonometric Functions and Trig Equations
Domain 1, 1 Range
0 lt y lt 1, solutions in QI and QII. 1 lt y lt 0,
solutions in QIII and QIV.
Domain Range
Domain 1, 1 Range 0, p
0 lt y lt 1, solutions in QI and QIV. 1lt y lt 0,
solutions in QII and QIII.
0 lt y lt 1, solutions in QI and QIII. 1 lt y lt 0,
solutions in QII and QIV.
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Trigonometric IdentitiesSummation Difference
Formulas
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Trigonometric IdentitiesDouble Angle Formulas
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Trigonometric IdentitiesHalf Angle Formulas
The quadrant of
determines the sign.
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Law of Sines Law of Cosines
Law of sines
Law of cosines
Use when you have a complete ratio SSA.
Use when you have SAS, SSS.
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Vectors

• A vector is an object that has a magnitude and a
  direction.
• Given two points P1 and P2 on
  the plane, a vector v that connects the points
  from P1 to P2 is
• v i j.
• Unit vectors are vectors of length 1.
• i is the unit vector in the x direction.
• j is the unit vector in the y direction.
• A unit vector in the direction of v is v/v
• A vector v can be represented in component form
• by v vxi vyj.
• The magnitude of v is v
• Using the angle that the vector makes with x-axis
  in standard position and the vectors magnitude,
  component form can be written as v vcos(?)i
  vsin(?)j

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Vector Operations
Scalar multiplication A vector can be multiplied
by any scalar (or number). Example Let v 5i
4j, k 7. Then kv 7(5i 4j) 35i 28j.
Dot Product Multiplication of two vectors. Let v
vxi vyj, w wxi wyj. v w vxwx vywy
Example Let v 5i 4j, w 2i 3j. v w
(5)(2) (4)(3) 10 12 2.
Alternate Dot Product formula v w
vwcos(?). The angle ? is the angle
between the two vectors.
v
?
w
Two vectors v and w are orthogonal
(perpendicular) iff v w 0.
Addition/subtraction of vectors Add/subtract
same components. Example Let v 5i 4j, w 2i
3j. v w (5i 4j) (2i 3j) (5 2)i
(4 3)j 3i 7j. 3v 2w 3(5i 4j)
2(2i 3j) (15i 12j) (4i 6j) 19i
6j. 3v 2w
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Acknowledgements

• Unit Circle http//www.davidhardison.com/math/tri
  g/unit_circle.gif
• Text Blitzer, Precalculus Essentials, Pearson
  Publishing, 2006.