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:

�

s = rθ (

�

θ must be in radians)

Angular Speed (velocity): ω = central angletime

= θt

Linear Speed (velocity): ν = arc lengthtime

= st= rθt= rω

�

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

�

−1≤ x ≤1

�

− π2 ≤ y ≤

π2

arccosine x

�

−1≤ x ≤1

�

0 ≤ x ≤ π arctangent x

�

−∞ ≤ x ≤ ∞

�

− π2 ≤ y ≤

π2

Simple Harmonic Motion: Equation: d = asinω t or d = acosω t where: d = distance from origin, t = time,

�

a = amplitude, period = 2πω , frequency = ω2π

Bearings Trigonometry: angle

�

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

�

θ = 90˚ + 62˚ = 152˚ S 20˚ W : 20˚ west of due south or

�

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

�

v = g θ( ) . Pythagorean:

�

sin2 u + cos2 u =1

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1+ tan2 u = sec2 u

�

1+ cot2 u = csc2 u and

�

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:

�

sin −u( ) = −sinu

�

tan −u( ) = −tanu

�

cos −u( ) = cosu

�

csc −u( ) = −cscu cot −u( ) = − cotu

�

sec −u( ) = secu

Chapter 7Analytic Trigonometry Sum and Difference Formulas:

�

sin u + v( ) = sinucosv + cosusinv

�

cos u + v( ) = cosucosv − sinusinv

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sin u − v( ) = sinucosv − cosusinv

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cos u − v( ) = cosucosv + sinusinv

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tan u + v( ) = tanu + tanv1− tanutanv

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tan u − v( ) = tanu − tanv1+ tanutanv

Double Angle Formulas:

�

sin2u = 2sinucosu

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cos2u = cos2 u − sin2 u

�

tan2u = 2tanu1− tan2 u

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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:

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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( )

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r = a2 + b2 and

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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:

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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:

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n!=1⋅ 2 ⋅ 3 ⋅ 4! n −1( ) ⋅ n Fibonacci Sequence (a recursive sequence)

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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):

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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

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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

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E2 can occur in

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m2 ways after

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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

�

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:

�

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!

⎛ ⎝ ⎜

⎞ ⎠ ⎟