MTH112 Trigonometry
Definitions: sinθ = opp.hyp.
= yr
cscθ = 1sinθ
= ry
cosθ = adj.hyp.
= xr
secθ = 1cosθ
= rx
tanθ = opp.adj.
= sinθcosθ
= yx
cotθ = 1tanθ
= cosθsinθ
= xy
Radian: Measure of the central angle
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θ that intercepts an arc s equal in length to the radius r. Full circle =
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2π radians. π radians = 180° Arc Length:
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s = rθ (
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θ must be in radians)
Angular Speed (velocity): ω = central angletime
= θt
Linear Speed (velocity): ν = arc lengthtime
= st= rθt= rω
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π6
�
π4
�
π3
�
π2
�
2π3
�
3π4
�
7π6
�
5π6
�
5π4
�
4π3
�
π
�
5π3
�
7π4
�
11π6
0˚360˚
�
2π
�
0180˚
90˚
30˚
45˚
60˚120˚
135˚
150˚
210˚
225˚
240˚ 300˚
315˚
330˚
x
y
�
1, 0( )�
32,12
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
22,22
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
12,32
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
−12,32
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
−22,22
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
−32,12
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
−12,−
32
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
−22,−
22
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ �
−32,−12
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
12,−
32
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
22,−
22
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
32,−12
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
θ
s = rθr
r
θ
s = rr
r
Graphs of Sine and Cosine Functions: For the functions y = d + asinb x − c( ) and y = d + acosb x − c( ) Wave base line : y = d Amplitude = a Max: y = d + a Min: y = d – a Period = 2πb Phase Shift = c Angle: θ = b x − c( ) Graphs of Tangent Function: For the functions y = d + a tanb x − c( ) Base line for inflection point: y = d, Period = π
b , Vertical asymptotes for tangent: b x − c( ) = π
2 + nπ ; When the angle is straight up or down Inverse Functions: Domain Range arcsine x
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−1≤ x ≤1
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− π2 ≤ y ≤
π2
arccosine x
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−1≤ x ≤1
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0 ≤ x ≤ π arctangent x
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−∞ ≤ x ≤ ∞
�
− π2 ≤ y ≤
π2
Simple Harmonic Motion: Equation: d = asinω t or d = acosω t where: d = distance from origin, t = time,
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a = amplitude, period = 2πω , frequency = ω2π
Bearings Trigonometry: angle
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θ is measured counter-clockwise from the positive x-axis Ocean Navigation and Surveying: Direction given as an acute angle east or west of a north or south reference line E.g. N 62˚ W: 62˚ west of due north or
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θ = 90˚ + 62˚ = 152˚ S 20˚ W : 20˚ west of due south or
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θ = 180˚ + 70˚ = 250˚ (or –90˚–20˚= –110˚) Aircraft Navigation: Direction given as an angle measured in degrees clockwise from due north E.g. Due north is 0˚ (simpler than 360˚ which would be the same direction) Due east would be a course of 90˚, south = 180˚, west = 270˚ Identities Note: u and/or v may also represent functions of an angle:
�
u = f θ( ) and
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v = g θ( ) . Pythagorean:
�
sin2 u + cos2 u =1
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1+ tan2 u = sec2 u
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1+ cot2 u = csc2 u and
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sinu = ± 1− cos2 u . . . etc. Cofunction Identities:
�
sin π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = cosu
�
cos π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = sinu
�
tan π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = cot u
�
cot π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = tanu
�
sec π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = cscu
�
csc π2− u
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = secu
Odd Functions: Even Functions:
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sin −u( ) = −sinu
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tan −u( ) = −tanu
�
cos −u( ) = cosu
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csc −u( ) = −cscu cot −u( ) = − cotu
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sec −u( ) = secu
Chapter 7Analytic Trigonometry Sum and Difference Formulas:
�
sin u + v( ) = sinucosv + cosusinv
�
cos u + v( ) = cosucosv − sinusinv
�
sin u − v( ) = sinucosv − cosusinv
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cos u − v( ) = cosucosv + sinusinv
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tan u + v( ) = tanu + tanv1− tanutanv
�
tan u − v( ) = tanu − tanv1+ tanutanv
Double Angle Formulas:
�
sin2u = 2sinucosu
�
cos2u = cos2 u − sin2 u
�
tan2u = 2tanu1− tan2 u
�
cos2u = 2cos2 u −1
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cos2u =1− 2sin2 u Half-Angle Formulas:
�
sin u2
= ± 1− cosu2
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cos u2
= ± 1+ cosu2
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tan u2
= sinu1+ cosu
= 1− cosusinu
Power-Reducing Formulas:
�
sin2 u = 1− cos2u2
�
cos2 u = 1+ cos2u2
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tan2 u = 1− cos2u1+ cos2u
Sum to Product Formulas:
�
sinu + sinv = 2sin u + v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟ cos
u − v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
�
cosu + cosv = 2cos u + v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟ cos
u − v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
�
sinu − sinv = 2cos u + v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟ sin
u − v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
�
cosu − cosv = −2sin u + v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟ sin
u − v2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Product to Sum Formulas:
�
sinu ⋅ sinv = 12 cos u − v( ) − cos u + v( )[ ]
�
sinu ⋅ cosv = 12 sin u + v( ) + sin u − v( )[ ]
�
cosu ⋅ cosv = 12 cos u − v( ) + cos u + v( )[ ]
�
cosu ⋅ sinv = 12 sin u + v( ) − sin u − v( )[ ]
Chapter 8: Solving Triangles In any triangle rABC, with angles A, B, C, and sides a, b, c Law of Sines:
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asinA
= bsinB
= csinC
Law of Cosines:
�
a2 = b2 + c2 − 2bc cosA
�
b2 = a2 + c2 − 2ac cosB
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c2 = a2 + b2 − 2abcosC Area of a Triangle
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Area = 12 bh
In an Oblique Triangle:
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Area = 12 bc sinA = 1
2 absinC = 12 ac sinB
Heron’s Area Formula:
�
Area = s s− a( ) s− b( ) s− c( ) where
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s = a + b + c2
Vectors Vector arithmetic with components: For
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! u = u1,u2 and
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! v = v1,v2
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! u + ! v = u1 + v1, u2 + v2
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! u − ! v = u1 − v1, u2 − v2
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k! u = ku1,ku2
Magnitude:
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! u = u1( )2 + u2( )2 Dot Product:
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! u ⋅ ! v = u1v1 + u2v2
Angle between vectors:
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cosθ =! u ⋅ ! v ! u ! v
Work:
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W =! F ⋅! D
Changing Rectangular/Polar Forms of Vectors: Rectangular:
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a,b , a is horizontal (x) component, b is the vertical (y) component Polar:
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r,θ( ), r is magnitude (length),
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θ angle clock-wise from positive x-axis
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r = a2 + b2
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θ = tan−1 ba
⎛ ⎝ ⎜
⎞ ⎠ ⎟ (check signs of a, b for quadrant)
�
a = rcosθ
�
b = rsinθ
A
B
C
a
b
c
Vector Forms of Complex Numbers
Quadratic Formula For
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ax2 + bx + c = 0,
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x = −b ± b2 − 4ac2a
Converting between rectangular and trigonometric (polar) forms: For complex number z:
Standard or rectangular form:
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z = a + bi. Polar or trigonometric form:
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z = r cosθ + isinθ( ) with
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θ in 0,2π[ ) or 0˚,360˚[ ) .
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z = a + bi = r cosθ + isinθ( )
�
r = a2 + b2 and
�
θ = tan−1 ba
a = r cosθ and b = r sinθ Absolute Value (Magnitude) of a Complex Number z = a + bi = r = a2 + b2 Multiplication of Complex Numbers in Trigonometric Form
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r1 cosθ1 + isinθ1( )[ ] r2 cosθ2 + isinθ2( )[ ] = r1r2 cos θ1 + θ2( ) + isin θ1 + θ2( )[ ] Division of Complex Numbers in Trigonometric Form
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r1 cosθ1 + isinθ1( )r2 cosθ2 + isinθ2( )
= r1r2cos θ1 −θ2( ) + isin θ1 −θ2( )[ ]
DeMoivre's Theorem (Powers of Complex Numbers):
�
z n = r cosθ + i sinθ( )[ ]n = r n cos nθ + i sinnθ( ) Roots of Complex Numbers
�
zn = r cosθ + isinθ( )n
�
= rn cosθ + 2π kn
+ isinθ + 2π kn
⎛ ⎝ ⎜
⎞ ⎠ ⎟ where
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k = 0,1, 2,!, n −1
Chapter 11: Sequences, Series, and Probability 11.1 Sequences and Series Infinite Sequence: A function whose domain is the set of positive integers.
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a1,a2,a3,a4,!,an,! Where the terms of the sequence are:
�
a1 = f 1( ),
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a2 = f 2( ),
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a3 = f 3( ),
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a4 = f 4( ) , etc. Summation Notation: The sum of the first n terms of a sequence
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aii=1
n
∑ = a1 + a2 + a3 +!+ an
Factorial:
�
n!=1⋅ 2 ⋅ 3 ⋅ 4! n −1( ) ⋅ n Fibonacci Sequence (a recursive sequence)
�
a1 =1,
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a2 = 2 , . . . ,
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ak = ak−2 + ak−1 Terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… Properties of Summations: (c is a constant,
�
an and
�
bn are nth terms of different series)
1.
�
ci=1
n
∑ = cn 2.
�
caii=1
n
∑ = c aii=1
n
∑
3.
�
ai + bi( )i=1
n
∑ = aii=1
n
∑ + bii=1
n
∑ 4.
�
ai − bi( )i=1
n
∑ = aii=1
n
∑ − bii=1
n
∑
Series: Summation of sequences
Finite Series (nth partial sum of the sequence)
�
aii=1
n
∑ = a1 + a2 + a3 +!+ an
Infinite Series (sum all terms of an infinite sequence)
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aii=1
∞
∑ = a1 + a2 + a3 +!+ ai +!
11.2 Arithmetic Sequences and Partial Sums Arithmetic Sequence: Consecutive terms have the same difference: a2 − a1 = a3 − a2 =!= an − an−1 = d Test: Subtract terms. There is a common difference between each term: an − an−1 = d nth term of the sequence: an = a1 + d n −1( ) a1= first term of the sequence d = common difference n = ordinal number of the term to be found Terms: First a1 + d(1−1) = a1 + d(0) = a1 Second a2 = a1 + d(2 −1) = a1 + d Third a3 = a1 + d(3−1) = a1 + 2d Fourth a4 = a1 + d(4 −1) = a1 + 3d , etc. Note: Since the sequence could start with any number, a1 , the formula’s second part has
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d(n −1) . That means for the first term, n = 1, the second part is zero leaving just a1 .
Sum of a Finite Arithmetic Sequence (author’s formula):
�
Sn = n2a1 + an( ) = a1 + d n −1( )⎡⎣ ⎤⎦
1
n
∑
n = number of terms to sum a1= first term d = common difference
Sum of a Finite Arithmetic Sequence (alternate): Sn = a1 + d n −1( )⎡⎣ ⎤⎦1
n
∑ =n22a1 + d n −1( )⎡⎣ ⎤⎦
11.3 Geometric Sequences and Series Geometric Sequence: Terms in the sequence have a constant ratio
a2a1
=a3a2
=a4a3
=! = r
nth term in the sequence is in the form:
�
an = a1 rn−1
�
a1 = first term of the sequence ( note:
�
a1 r1−1 = a1 r
0 = a1 ⋅1= a1) r = common ratio (which is the base of the exponential) n = ordinal number of the term to be found
Geometric Series: Sum of a Finite Geometric Sequence
�
a1 ri−1
i=1
n
∑ = a11− rn
1− r
⎛
⎝ ⎜
⎞
⎠ ⎟
n = number of terms to be summed
�
a1 = first term of the sequence (
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a1 r0 = a1 ⋅1= a1)
r = common ratio
�
r ≠1( )
Sum of an Infinite Geometric Series (
�
r <1)
�
a1 ri
i=0
∞
∑ = a11− r
11.6 Counting Principles (Combinatorics) Fundamental Counting Principle: If event
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E1 can occur in
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m1 different ways and event
�
E2 can occur in
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m2 ways after
�
E1 has occurred, the total number of ways the two events can occur is
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m1 ⋅m2 . Permutations of n Elements: Selecting subsets of a group of n items where the order of selection matters (using three letters, ABC is different than BCA). The number of different permutations (different orderings) of n things is
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n!. Permutation of n Elements Taken r at a Time: (Note: You are selecting r elements and the
order selected matters.)
�
n Pr = n!n − r( )!
Distinguishable Permutations: (Note: You are selecting from a pool of n items, some of which are identical. Again, order of selection matters.) A set of n objects has k different types of items where
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n1 is the number of one type,
�
n2 the number of the second type, and so on such that:
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n = n1 + n2 + n3 +!+ nk.
Distinguishable Permutations =
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n!n1!⋅n2!⋅n3!⋅ ! ⋅ nk!
Combinations: Selecting subsets of a group of n items where order does not matter. (e.g. select three letters — ABC is the same as BCA because the same letters were selected)
Combinations of n Elements Taken r at a Time:
�
nCr = n!n − r( )!r!
= n Prr!
⎛ ⎝ ⎜
⎞ ⎠ ⎟
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