3 sequence and trigonometry
Sequences &Trigonometry 2016
III. Sequences and Trigonometry
PART I: Sequences
Words
Sequence
Progression
Arithmetic sequence
Geometric
Difference
Ratio
Series
Finite
Infinite
Term
First Term
Middle Term
Consecutive Terms
Partial sum
Pronunciation
/'si:kwəns/
Indonesian
Barisan
/ə'rɪθmətɪk 'si:kwəns/
Barisan Aritmatika
/'dɪfrəns/
Beda
/'prə'gre∫n/
Barisan
/dƷɪə'mətrɪk 'si:kwəns/
Barisan Geometri
/'reɪ∫ɪəʊ/
Rasio
/'faɪnaɪt/
Berhingga
/'sɪəri:z/
Deret
/'ɪnfɪnət/
Tak Berhingga
/fɜ:st tɜ:m/
Suku Pertama
/tɜ:m/
Suku
/'mɪdl tɜ:m/
Suku Tengah
/kən'sekjʊtɪv tɜ:mz/
/'pɑ:∫l sΛm/
Suku Berurutan
Jumlah sebagian
(suku)
Example:
•
An arithmetic sequence is a sequence of the form a, a+d, a+2d, ….
•
The number a is the first term, and d is the difference of the sequence.
•
The formula for n-th term of the arithmetic sequence is a + (n-1) d
•
a1, a2, a3 are the first three terms.
•
a(n+1)/2 is called middle term.
•
an and an+1 are two consecutive terms.
•
For arithmetic sequence a, a+d, a+2d, …, the formula for n-th partial
sum Sn is (n/2)(2a + (n − 1)d)
•
A geometric sequence is sequence in the form of a, ar, ar2, … where r is
the ratio.
•
The sum of finite or infinite sequence, a1+a2+ ..., is called series.
1 Sequences &Trigonometry
Sequences &Trigonometry 2016
Practice
Complete the sentences or give short answers.
1. The first 6 terms of the arithmetic sequence 13,7,1,... are ______________.
2. If the 10th term of an arithmetic sequence is 55 and the 2nd term is 7,
then its 1st term is ___.
3. The sum of the first 40 terms of the arithmetic sequence 3,7,11,15.., is
_____.
4. If the third term of a geometric sequence is 20 and the sixth term is 2.5,
then the its common ratio is ______.
5. The ___________________of a geometric sequence whose first term is 2
and ____________ is five are 2, 10, 50, 250, and 1250.
6. The ____________________ of sequence 1,2,4,8,... is 256.
7. If the 3rd term of a geometric sequence is 63/4, and the 6th term is
1701/32, then the 4th term is _________.
8. The sum of 1, 0.5, 0.25, 0.125, .. is ______________.
PART II: Greek Alphabet
Big Letters Small Letters Words
Α
α
alpha
Β
β
beta
Γ
γ
gamma
Δ
δ
delta
Ε
ε
epsilon
Ζ
ζ
zeta
Η
η
eta
Θ
θ
theta
Ι
ι
iota
Κ
κ
kappa
Λ
λ
lamda
Μ
μ
mu
Ν
ν
nu
Ξ
ξ
xi
Ο
ο
Π
π
Pronunciation
/'ælfə/
/'bi:tə/
/'gæmə/
/'deltə/
/'epsilən/
/'ziːtə/
/'iːtə/
/'θiːtə/
/aɪ'əʊtə/
/'kæpə/
/'læmdə/
/'mjuː/
/'njuː/
/'ksaɪ/
omicron /'əʊmɪkrən/
pi
/'paɪ/
2 Sequences &Trigonometry
Sequences &Trigonometry 2016
Ρ
ρ
rh o
Σ
σ
Τ
τ
tau
Υ
υ
upsilon
Φ
φ
phi
Χ
χ
chi
Ψ
ψ
ps i
Ω
ω
omega
sigma
/'rəʊ/
/'sɪgmə/
/'tɑʊ/
/'jʊpsɪlən/
/'faɪ/
/'kaɪ/
/'psaɪ/
/'əʊmɪgə/
PART III: Trigonometry
Words
Trigonometry
Adjacent side
Hypotenuse
Opposite side
Angle
Pronunciation
/trɪgə'nɒmətrɪ/
/ə'dƷeɪsnt saɪd/
Sisi samping
/ɒpəzɪt saɪd/
Sisi depan
/dɪ'gri:z/
Derajat
/'kwɒdrənt/
Kuadran
/rɪ'siprəkl/
Kebalikan
Sisi miring
/haɪ'pɒtənyu:z/
Sudut
/'æŋgl/
Degrees
Period
Periode
/'pɪərɪəd/
Quadrant
Radian
Radian
/'rəɪdɪən/
Reciprocal
Indonesian
Trigonometri
Trigonometric Function
sin
sine
/saɪn/
cos
cos; cosine
/kɒz/;/kɒzaɪn/
tan
se c
csc
tan; tangent
sec;
cosec;
/tæn/;/tændƷənt/
/s e k/
cot
cotangent
/'kəʊtændƷənt /
/'kəʊsek/
Examples:
¬ Trigonometry is the study of angle measurement.
¬ The hypotenuse will always be the longest side, and opposite from the
right angle.
3 Sequences &Trigonometry
Sequences &Trigonometry 2016
¬ The opposite side is the side directly across from the angle you are
considering.
¬ The adjacent side is the side next to angle you are considering.
¬ All functions have positive values for angles in Quadrant I, but only sine
has positive values for angles in Quadrant II.
¬ The sine and cosine functions have the period 2π (3600).
Practice
Fill the table below and say the equation, such as sin(00)=….
Degrees
Radians
0
0
30
π/6
45
π/4
60
π/3
90
π/2
Sin
Cos
Tan
Practice
Read the following expressions.
1. (Formulas for Addition and Subtraction)
sin(A + B) = sin A cos B + cos A sin B
cos(A - B) = cos A cos B + sin A sin B
tan(A + B) = tan A + tan B / (1 - tan A tan B)
2. (Phytagorean Identities)
sin2 θ + cos 2 θ= 1
tan2 θ +=
1 sec 2 θ
3. (Formula for Double Angle)
sin 2A = 2 sin A cos A
4. (Formula for half angle)
cos
5. (Cosine Rule)
a² = b² + c² - 2bc cos A
4 Sequences &Trigonometry
θ
1 + cos θ
= ±
2
2
Sequences &Trigonometry 2016
Exercise
Complete the sentences or give short answers.
1. Tangent has positive values for angles in _______________, and
______________ has positive values for angles in Quadrant IV.
2. The tangent and cotangent functions have the period __________
3. If sin α = 0.8, then the value of sin (180-α) is _________ and the value
of tan α is ______________
4. Without using calculator, find cos(150).
5. Find the values of x for which sin 3x = 0.5 if it is given that 0
III. Sequences and Trigonometry
PART I: Sequences
Words
Sequence
Progression
Arithmetic sequence
Geometric
Difference
Ratio
Series
Finite
Infinite
Term
First Term
Middle Term
Consecutive Terms
Partial sum
Pronunciation
/'si:kwəns/
Indonesian
Barisan
/ə'rɪθmətɪk 'si:kwəns/
Barisan Aritmatika
/'dɪfrəns/
Beda
/'prə'gre∫n/
Barisan
/dƷɪə'mətrɪk 'si:kwəns/
Barisan Geometri
/'reɪ∫ɪəʊ/
Rasio
/'faɪnaɪt/
Berhingga
/'sɪəri:z/
Deret
/'ɪnfɪnət/
Tak Berhingga
/fɜ:st tɜ:m/
Suku Pertama
/tɜ:m/
Suku
/'mɪdl tɜ:m/
Suku Tengah
/kən'sekjʊtɪv tɜ:mz/
/'pɑ:∫l sΛm/
Suku Berurutan
Jumlah sebagian
(suku)
Example:
•
An arithmetic sequence is a sequence of the form a, a+d, a+2d, ….
•
The number a is the first term, and d is the difference of the sequence.
•
The formula for n-th term of the arithmetic sequence is a + (n-1) d
•
a1, a2, a3 are the first three terms.
•
a(n+1)/2 is called middle term.
•
an and an+1 are two consecutive terms.
•
For arithmetic sequence a, a+d, a+2d, …, the formula for n-th partial
sum Sn is (n/2)(2a + (n − 1)d)
•
A geometric sequence is sequence in the form of a, ar, ar2, … where r is
the ratio.
•
The sum of finite or infinite sequence, a1+a2+ ..., is called series.
1 Sequences &Trigonometry
Sequences &Trigonometry 2016
Practice
Complete the sentences or give short answers.
1. The first 6 terms of the arithmetic sequence 13,7,1,... are ______________.
2. If the 10th term of an arithmetic sequence is 55 and the 2nd term is 7,
then its 1st term is ___.
3. The sum of the first 40 terms of the arithmetic sequence 3,7,11,15.., is
_____.
4. If the third term of a geometric sequence is 20 and the sixth term is 2.5,
then the its common ratio is ______.
5. The ___________________of a geometric sequence whose first term is 2
and ____________ is five are 2, 10, 50, 250, and 1250.
6. The ____________________ of sequence 1,2,4,8,... is 256.
7. If the 3rd term of a geometric sequence is 63/4, and the 6th term is
1701/32, then the 4th term is _________.
8. The sum of 1, 0.5, 0.25, 0.125, .. is ______________.
PART II: Greek Alphabet
Big Letters Small Letters Words
Α
α
alpha
Β
β
beta
Γ
γ
gamma
Δ
δ
delta
Ε
ε
epsilon
Ζ
ζ
zeta
Η
η
eta
Θ
θ
theta
Ι
ι
iota
Κ
κ
kappa
Λ
λ
lamda
Μ
μ
mu
Ν
ν
nu
Ξ
ξ
xi
Ο
ο
Π
π
Pronunciation
/'ælfə/
/'bi:tə/
/'gæmə/
/'deltə/
/'epsilən/
/'ziːtə/
/'iːtə/
/'θiːtə/
/aɪ'əʊtə/
/'kæpə/
/'læmdə/
/'mjuː/
/'njuː/
/'ksaɪ/
omicron /'əʊmɪkrən/
pi
/'paɪ/
2 Sequences &Trigonometry
Sequences &Trigonometry 2016
Ρ
ρ
rh o
Σ
σ
Τ
τ
tau
Υ
υ
upsilon
Φ
φ
phi
Χ
χ
chi
Ψ
ψ
ps i
Ω
ω
omega
sigma
/'rəʊ/
/'sɪgmə/
/'tɑʊ/
/'jʊpsɪlən/
/'faɪ/
/'kaɪ/
/'psaɪ/
/'əʊmɪgə/
PART III: Trigonometry
Words
Trigonometry
Adjacent side
Hypotenuse
Opposite side
Angle
Pronunciation
/trɪgə'nɒmətrɪ/
/ə'dƷeɪsnt saɪd/
Sisi samping
/ɒpəzɪt saɪd/
Sisi depan
/dɪ'gri:z/
Derajat
/'kwɒdrənt/
Kuadran
/rɪ'siprəkl/
Kebalikan
Sisi miring
/haɪ'pɒtənyu:z/
Sudut
/'æŋgl/
Degrees
Period
Periode
/'pɪərɪəd/
Quadrant
Radian
Radian
/'rəɪdɪən/
Reciprocal
Indonesian
Trigonometri
Trigonometric Function
sin
sine
/saɪn/
cos
cos; cosine
/kɒz/;/kɒzaɪn/
tan
se c
csc
tan; tangent
sec;
cosec;
/tæn/;/tændƷənt/
/s e k/
cot
cotangent
/'kəʊtændƷənt /
/'kəʊsek/
Examples:
¬ Trigonometry is the study of angle measurement.
¬ The hypotenuse will always be the longest side, and opposite from the
right angle.
3 Sequences &Trigonometry
Sequences &Trigonometry 2016
¬ The opposite side is the side directly across from the angle you are
considering.
¬ The adjacent side is the side next to angle you are considering.
¬ All functions have positive values for angles in Quadrant I, but only sine
has positive values for angles in Quadrant II.
¬ The sine and cosine functions have the period 2π (3600).
Practice
Fill the table below and say the equation, such as sin(00)=….
Degrees
Radians
0
0
30
π/6
45
π/4
60
π/3
90
π/2
Sin
Cos
Tan
Practice
Read the following expressions.
1. (Formulas for Addition and Subtraction)
sin(A + B) = sin A cos B + cos A sin B
cos(A - B) = cos A cos B + sin A sin B
tan(A + B) = tan A + tan B / (1 - tan A tan B)
2. (Phytagorean Identities)
sin2 θ + cos 2 θ= 1
tan2 θ +=
1 sec 2 θ
3. (Formula for Double Angle)
sin 2A = 2 sin A cos A
4. (Formula for half angle)
cos
5. (Cosine Rule)
a² = b² + c² - 2bc cos A
4 Sequences &Trigonometry
θ
1 + cos θ
= ±
2
2
Sequences &Trigonometry 2016
Exercise
Complete the sentences or give short answers.
1. Tangent has positive values for angles in _______________, and
______________ has positive values for angles in Quadrant IV.
2. The tangent and cotangent functions have the period __________
3. If sin α = 0.8, then the value of sin (180-α) is _________ and the value
of tan α is ______________
4. Without using calculator, find cos(150).
5. Find the values of x for which sin 3x = 0.5 if it is given that 0