Mathematics A. Algebra a. Basic Axioms of Algebra Let a, b and c be any real number. 1. Closure Axiom for Addition, 2. Closure Axiom for Multiplication, b. Basic Law of Natural Numbers Let a, b and c be any number. 1. Commutative Law for Addition, 2. Associative Law for Addition, 3. Commutative Law for Multiplication, 4. Associative Law for Multiplication, 5. Distributive Law,
a+b=c ab = c a+b=b+a a + (b + c) = (a + b) + c axb=bxa a(bc) = (ab)c a(b + c) = ab + ac
c. Basic Property of Algebra Let ‘a’ be any number. 1. Additive Identity Property, 2. Additive Inverse Property, 3. Multiplicative Identity Property, 4. Multiplicative Inverse Property,
a+0=a a + (-a) = 0 ax1=a a(1/a) = 1 where a ≠ 0
d. Basic Laws of Equality Let a, b, c and d be any number. 1. Reflexive Property, 2. Symmetric Property, 3. Transitive Property, 4. If a = b and c = d, then a + c = b + d. 5. If a = b and c = d, then ac = bd.
a=a If a = b then b = a If a = b and b = c, then a = c
e. Inequality – a statement that one quantity is greater than or less than the other quantity. 1. Symbols used in Inequality i. ii.
> <
is greater than is less than
≥
iii.
equal to ≤
iv.
is greater than or is less than or equal to
2. Theorems on Inequality i. ii. iii. iv.
a > b if and only if -a < -b If a > 0, then -a < 0 If -a < 0, then a > 0 If a > b and c < 0, v. then ac < bc vi. If a > b and c > d, vii. then (a + c) > (b + d)
viii. ix.
xi.
If a > b, c > d and a, b , c, d > 0, then ac > bd If a > 0, b > 0 and a > b, 1 1 x. then a > b
xii. f. Effects of Zero and Negative xiii. Let a ≠ 0. 1. a x 0 = 0 0 2. a =0 3.
a 0
= undefined
4.
a ∞
=0
5. -1(a) = -a 6. 7. -1(-a) = a 8. -1(a + b) = -a – b 9. -1(a – b) = -a + b 10. a(-b) = (-a)(b) = -(ab) 11. (-a)(-b) = ab
g. Laws of Exponent (Index Law) 1.
an =an 1 x an 2 x a n3 x … an
2.
am x an=am+ n m
m n
n
7.
a =√ am
8.
1 ∧1 am −m a = −m =am a a0 =1 provided a ≠ 0
3.
a =a m−n n a
4.
( a m ) =a mn
9.
5.
( abc )m=am bm c m
10. If a =a
6.
n
m
n
, then m=n
provided a ≠ 0
a m am = m b b
()
11. h. Properties of Radicals 1.
1 n
n
a =√ a m n
n
m
2.
a =√ am=( √n a )
3.
( √n a ) =√n an=a
4.
√n a x √n b=√n ab
5.
√n a = n a √n b b
n
√
provided b
≠ 0
i. j. Properties of Logarithm 1.
log a MN =log a M + log a N
2.
log a
M =log a M −log a N N
n
3.
log a M =n log a M
4.
log a a=1
x
5.
log a a =x log a a=x
6.
log a 1=0
7. If log a M =N , then N
a =M 8. If log a M=log a N 9. then M =N
17.
x=antilo g b a x 18. a =antilo g a x
19. log 10 1250=log 10 ( 1000 x 1.25 ) 20. log 10 1250=log 1000+ log 1.25
10. Napierian Logarithm log e M =ln M 11. e=2.718281828 12. 13. Common Logarithm log 10 M =log M 14.
log M ln M 15. log N M = log N = ln N 16. If log b x=a ,
then
21. log 10 1250=3+0.09691 22. 23. Where: 24. 3 , the integral part is called the characteristic 25. 0.09691, the nonnegative decimal fraction part is called mantissa 26.
27. 28. k. Polynimials 1. Expanding Brackets – by multiplying two brackets together, each term in one bracket is multiplied by each term of the other bracket. 29. (a + b + c)(x + y) = ax + ay + bx + by + cx + cy 2. Factorization – opposite process of expanding brackets. 30. 2x2 – 6x + 4 = 2(x – 2)(x – 1) 3. Special Products and Factoring i. (x + y)(x – y) = x2 – y2 ii. (x + y)2 = x2 +2xy + y2 iii. (x - y)2 = x2 – 2xy + y2 iv. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz v. x3 + y3 = (x + y)(x2 – xy +y2) vi. x3 – y3 = (x – y)(x2 +xy + y2) vii. xn + yn = (x + y)(xn – 1 – xn – 2y + xn – 3y2 – xn – 4y3 +… yn – 1) viii. xn – yn = (x – y)(xn – 1 + xn – 2y + xn – 3y2 + xn – 4y3 +… yn – 1) 4. Division of Polynomials i.
By Long Division
ii.
Synthetic Division
5. Factor Theorem – Consider a function f(x). If f(a) = 0, then (x – a) is a factor of f(x). 6. Remainder Theorem – if a polynomial f(x) is divided by (x – r) until a remainder which is free of x is obtained, the remainder is f(r). If f(r) = 0 the (x – r) is a factor of f(x).
7. Binomial Theorem - (x + y)n i. Properties The number of terms in the expansion is n + 1. The first exponent xn & the last term is yn. The exponent x descends linearly from n to zero. The exponent of y ascends linearly from zero to n. The sum of the exponents of x & y in any of the terms is equal to n. The coefficient of the second term and the second from the last term is n. ii. Pascal’s Triangle – used to determine the coefficients of the terms in a binomial expansion. n r th term of ( x+ y ) iii. th
r =
iii.
n! n−r+1 r−1 x y ( n−r +1 ) ! ( r −1 ) !
n To get the middle term (for even value of n). Set r= 2 +1
iv.
iv. Coefficient of Next Term (Coefficient of previous term)(exponent of x ) C= v. ( exponent of y ) +1 v.
Sum of Coefficient of Variables – Substitute unity (1) to each variables. If ( x+ a )n , subtract the value of an . vi. 2
8. Quadratic Formula – For quadratic equation a x + bx+ c=0 vii.
x=
−b ± √ b2−4 ac 2a
i. Sum of Roots,
x 1+ x 2=−b /a
ii. Product of Roots,
x 1 x 2=c /a
iii. l. Partial Fractions – Functions of x that can be expressed in the form of P(x)/Q(x), where both P(x) and Q(x) are polynomials of x, is known as rational functions. 1. Improper Functions – if the degree of P(x) is ≥ degree of Q(x). 2. Proper Functions - if the degree of P(x) is < degree of Q(x). 3. Methods of Resolving Proper Fractions into Partial Fraction i. Case 1 – Factors of the denominator all linear, none repeated. 2 x2 +3 x−1 A B C = + + iv. ( x−1 )( x +2 )( x−3 ) x−1 x +2 x−3 v.
Multiply it by LCM and solve for A, B and C.
i. Case 2 – Factors of the denominator all linear, some repeated. 2 x 2 +2 x−1 A B C D = + + + 3 2 vi. ( x−1 )( x +2 ) x−1 x +2 ( x +2 ) ( x +2 )3 vii. Multiply it by LCM, expand and equate the coefficients of like powers to solve for A, B, C and D. i. Case 3 – Some factors of the denominator quadratic, none repeated. 2 x2 +2 x−1 A Bx+C Dx+ E = + 2 + 2 2 2 viii. ( x−1 )( x +2)( x +2 x +4) x−1 x +2 x + 2 x +4 ix. Multiply it by LCM, expand and equate the coefficients of like powers to solve for A, B, C, D and E. i. Case 4 – Some factors of the denominator quadratic, some repeated. 2 x 2+2 x−1 A Bx +C Dx+ E = + 2 + 2 2 x. x−1 x +2 ( x2 +2 ) 2 ( x−1 ) ( x +2 ) xi. Multiply it by LCM, expand and equate the coefficients of like powers to solve for A, B, C, D and E. xii. m. Proportion – is a statement of equality between to ratios. In the following proportion a c a :b=c : d∨ = xiii. b d xiv. xv. xvi. xvii.
b and c are called the means. a and d are the extremes d is the fourth proportional to a, b and c In the ratio a/b, a is the antecedent and b is the consequent.
1. Mean Proportional - the mean proportional between two terms a and b = 2. Properties of Proportion i.
Proportion by Inversion a c b d If = , then = ii. b d a c iii.
v.
Proportion by Alteration a c a b iv. If b = d , then c = c Proportion by Composition vi. a c a+b c +d If = , then = b d b d
√ ab .
vii.
Proportion by Division viii. a c a−b c−d If = , then = b d b d
ix.
Proportion by Composition and Division x. a c a+b c+ d If = , then = b d a−b c−d
xi. n. Variation – a mathematical function that relates the value of one variable to those of other varianles.
1. Direct Variation –
x ∝ y∨x=ky
2. Inverse Variation –
x ∝1 / y ∨x=k / y
3. Joint Variation –
x ∝ y /z 2∨x =ky / z 2
xii. o. Progressions 1. Arithmetic Progressions – a sequence of numbers in which the difference of any two adjacent terms is constant. an =a1 + ( n−1 ) d nth term of A . P . i. or an =am + ( n−m ) d ii.
∑ of n terms of A . P .
n S= ( a1 +an ) 2
or
S=
n [ 2a 1+ ( n−1 ) d ] 2
2. Geometric Progression – a sequence of numbers in which the ratio of any two adjacent terms is constant. n−1 n−m an =a1 r an =am r nth term of G . P. i. or ii.
iii.
∑ of n terms of G. P . xiii.
a1 ( r n−1 ) S= when r >1 r −1
xiv.
a1 ( 1−r n ) S= when r <1 1−r
∑ of Infinite G . P .
S=
a1 1−r
3. Harmonic Progression – a sequence of numbers in which their reciprocal forms an arithmetic progression. xv. p. Worded Problems Work Done=Rate x Time 1. Work Problem: Time 1 xvi. Rate= ¿ finish the work ¿ 2. Age Problem: 3. Digit Problem:
The difference of the ages of two persons is constant. For 3-digit number: 100h + 10t + u xvii. Let: h = hundred’s digit xviii. t = ten’s digit xix. u = unit’s digit
4. Number Problem 5. Clock Problem: i. If the minute hand moves a distance of x, the hour hand moves x/12. ii. If the second hand move a distance x, the minute hand moves x/60 and the hour hand moves x/720.
iii. In 12 hours, the minute-hand and the hour-hand of the clock overlap each other for 11 times. iv. Each five-minute mark is subtends an angle of 30° from the center of the clock. 6. Mixture Problem 7. Motion Problem (Uniform Motion or Constant Speed) xx. S = vt, S = Distance, v = Speed, t = time xxi. B. Probability and Statistics a. Techniques of Counting 1. Tree Diagram - This technique is a visual form of counting technique. 2. Multiplication Principle - This counting technique is used when a situation becomes somewhat complicated when we try to count the number of ways two or more events can occur in succession or in order. 3. Permutation – is an arrangement of the elements of a set in a definite in a definite order. xxii. Rules on Permutation i. Permutation of n Elements taken all at a Time nPr=n ! (if n=r) xxiii. ii.
Permutation of n Elements taken r at a Time n! nPr= (if r
iii. Permutation of n Elements with Some are Alike n! n Pr 1 r 1 rk = xxv. r1 ! r2 ! … . rk ! xxvi.
( r 1 ,r 2 , … . r k
are cells containing objects that are
of the same kind only) iv. Circular Permutation (one position must be fixed) xxvii. ( n−1 ) Pr ¿ ( n−1 ) ! 4. Combination – A combination is an arrangement of objects which does not involve the order of selection. n! nCr= xxviii. ( n−r ) ! r ! i.
Combination of n things taken 1, 2, 3… n at a time nCr=( 2n −1 ) ! xxix.
b. Probability – the chance of an event occurring. 1. A probability experiment is a chance process that leads to well-defined results called outcomes. 2. An outcome is the result of a single trial of a probability experiment. 3. A sample space, S, is the set of all possible outcomes of a probability experiment. 4. An event is a subset of a sample space.
5. The complement of an event A with respect to S is the subset of all elements of S that are not in A. the complement of A has the symbol A’. 6. The Probability of the Occurrence of an Event i. Single Event n ( E) P ( E ) = xxx. n (S ) ii.
Multiple Events Dependent and Independent Events – Two or more events are said to be dependent if the happening of one affects the probability of the others. And the independent if the happening of one does not affect the probability of the other. P ( E )=P ( E1 ) x P ( E2 ) x … P( E n) xxxi.
Mutually Exclusive Events – two or more events are said to be mutually exclusive if it is impossible for more than one of them to happen in a single trial xxxii. P ( E )=P ( E1 ) + P ( E2 ) +… P( E n) iii. Repeated Trials: The probability that an event can occur exactly r times in n trials is: r n−r xxxiii. P ( E )=nCr ( p ) ( q ) xxxiv. Where p is the probability that the event will happen and q is the probability that the event will fail. iv. The “Atleast one” Condition: The probability that an event can occur at least once in n trials is: P ( E )=1−Q xxxv. xxxvi. Where Q is the probability that the event will totally fail. v. Venn Diagram – A mathematical diagram representing sets as circles with their relationships to each other expressed through overlapping positions, so that all possible relationships between the sets are shown. c. Statistics - is the science that deals with the collection, organization, analysis, interpretation, and presentation of data. 1. Measures of Central Tendency of Ungrouped Data i. Arithmetic Mean - the quotient of the sum of the values and the total number of values. N x1 + x 2 + x 3+ …+ x N 1 x ´ = x ∨´ x = ∑ i xxxvii. N i N ii. Median - the midpoint of the data array. iii. Mode - the value that occurs most often in a data set. 2. Measures of Variation for Ungrouped Data i. Range: R = Highest observation – Lowest Observation ii. Mean Absolute Deviation
´ x∨¿ N M AD=¿
Σ∨x−
xxxviii. iii. Variance xxxix.
S 2N −1=
Σ ( x−´x )2 N −1
iv. Standard Deviation: For fx991-ES Plus (Mode Stat: Shift 1 > 4 > 4) s= √ S2N−1 xl. C. Advance Mathematics a. A matrix is a rectangular array of numbers/quantities arranged in rows and columns usually enclosed by a pair of brackets. A matrix is also denoted by a single capital letter. 1. Classification of Matrices i. Square Matrix – a matrix in which the number of rows equals the number of columns ii. Zero or Null Matrix – a matrix wherein all elements are zero. iii. Identity matrix – is a square matrix in which the diagonal elements are 1 (one) and all the off-diagonal elements are zero. iv. Row matrix – is a matrix having only one row and “n” columns. It is also called a row vector. v. Column matrix – is a matrix with “m” rows and only one column. It is also called a column vector. vi. Diagonal matrix – is a square matrix wherein all off-diagonal elements are zero. vii.Scalar matrix – is a square matrix for which all elements on the main diagonal are equal. viii. Symmetric matrix – is a square matrix wherein the elements about its main diagonal are symmetric (i.e. aij = aji). ix. Lower triangular matrix – is a square matrix whose elements above its principal diagonal are zero. x. Upper triangular matrix – is a square matrix whose elements below its principal diagonal are zero. xi. Triangular Matrix – has zeros in all positions above or below the diagonal. 2. Matrix Operations i. Equality. Two matrices A and B are said to be equal (A = B) if and only if they are of the same order, and each element of A is equal to the corresponding element of B. ii. Addition and Subtraction of Matrices. Two matrices A and B can be added (or subtracted) if they are of the same order. Their sum or difference is obtained by adding or subtracting their corresponding elements. iii. Multiplication of a Matrix by a Scalar. The product of a scalar K and a matrix A is obtained by multiplying each element of the matrix A by the scalar K. iv. Transpose of a Matrix. The transpose of matrix A, denoted by AT (or A’) is obtained by interchanging the rows and columns of A. The transpose of a symmetric matrix is equal to the matrix itself, i.e. AT = A.
v.
Multiplication of Matrices. The matrices A and B can be multiplied in the order AB if and only if the number of columns of A is equal to the number of rows of B. Such matrices are said to be conformable for multiplication. vi. Differentiation and Integration of Matrices. The derivative (or integral) of a matrix is obtained by differentiating (or integrating) each element of the matrix. vii. Conjugate of a Matrix. When A is a matrix having complex numbers as elements, the conjugate of A, denoted by Ā, is obtained by replacing each element by its conjugate. For the complex number a + bi, its conjugate is a – bi. viii. Inverse of a Square Matrix. The inverse of a square matrix A is defined as a matrix A–1 with elements of such magnitudes that the product of the original matrix A and its inverse A–1 equals an identity or unit matrix, I; that is, A A−1= A−1 A=I xli. xlii. xliii.
NOTE: Not every matrix has an inverse. Theorems on Inverse of Matrix −1 −1 −1 If A is non-singular matrix, then A is non-singular and ( A ) = A .
If A and B are non-singular matrices, then AB is non-singular and ( AB )−1=B−1 A−1 .
−1 −1 If A is non-singular matrix, then ( AT ) =(A ) T .
Two methods of finding the Inverse of a Square Matrix Gauss – Jordan Method Adjoint Method 3. Determinant of a Matrix. The determinant D, is a scalar calculated from a square matrix. i. Theorems on Determinants of Any Order Theorem 1: The number of terms in the expansion of a determinant of order n is n ! .
Theorem 2: If the corresponding rows and columns of a determinant are T T interchanged, its value is unchanged. | A|=| A |∨det A=det A .
Theorem 3: If any two columns or rows of a determinant are interchanged, the sign of the determinant is changed. Theorem 4: If all the elements in any two columns or rows of a determinant are zero, the value of the determinant is zero. Theorem 5: If any two columns or rows of a determinant have their corresponding elements identical or proportional, its value is zero. Theorem 6: If each element of a column or row in a determinant is multiplied by the same number k, the value of the determinant is multiplied by k. Theorem 7: If three determinant D1, D2, and D3 have corresponding elements equal, except for one column (or row) in which the element D1 are the sums of the corresponding elements of D2 and D3, then D1 = (D2 + D3).
ii.
Theorem 8: If each element of any column (or row) of a determinant is multiplied by the same number k and added to the corresponding elements of another column (or row), the value of the determinant is unchanged. Theorem 9: The value of the determinant is the algebraic sum of the products obtained by multiplied each element of a column (or row) by its cofactor or signed minor. Minors and Cofactors th th xliv. The minor M ij of the element aij in the i row and j column in any determinant of order n is that new determinant of order (n−1) formed from the elements remaining after deleting the i
th
row and
j
th
column. xlv. xlvi. The cofactor Aij of the element aij in any determinant of order n is that i+ j signed minor determined by A ij =(−1 ) M ij iii.
Evaluation of Determinants of Any Order
Pivotal Element Method Expansion by Minors /Cofactor Expansion Chio’s Method Dodgson’s Method of Condensation b. Complex Numbers xlvii. A complex number is of the form x+ iy (standard form) where x and y
are real numbers and i(i=√ −1) , which is called the imaginary unit. If
z=x +iy , then
imaginary part of
x is called the real part of
z
and
z and are denoted by ℜ( z ) and
y
is called the
ℑ(z) respectively.
The symbol z , which can stand for any of a set of complex numbers, is called a complex variable. 1. Operations of Complex Numbers i. Addition/Subtraction of Complex Numbers. Add/Subtract real part to real part 2 and imaginary to imaginary part. And applying i =−1 . ii. Multiplication of Complex Numbers. Similar to multiplication of polynomials. iii. Division of Complex Numbers. Multiply both numerator and denominator by the conjugate of the denominator. 2. Conjugate of Complex Numbers. Simply change the sign of the imaginary part. 3. Theorems on Complex Numbers. i. If x+ iy=0, then x=0∧ y=0 . ii. If x 1+i y 1=x 2+i y 2 , then x 1=x 2∧ y 1= y 2 .
iii. If ( x 1+i y 1 )( x2 +i y 2) =0, then one of the factors is zero. 4. Absolute Value. The absolute value or modulus of a complex number
z=x +iy is
|z|= √ x 2+ y 2 .
denoted by |z|.
5. Graphical Representation of Complex Numbers. Since a complex number z=x ++iy can be considered as an ordered pair of real numbers, we can represent a complex number by a point in an Diagram or z-plane. The xy-axis
xy− plane called the complex plane or Argand consist of two perpendicular axes; the horizontal
x-axis called the real axis and the vertical y-axis called the imaginary axis. 6. Polar or Trigonometric Form of Complex Numbers. xlviii. Consider complex number z=x +iy . x=rcosθ ; y=rsinθ xlix. z=r ( cos θ+i sin θ )∨z=rcisθ∨z=r ∠ θ
l. Substitute x and y; li. General Polar Form:
z=∠ ( θ+2 πk )
¿ θ+360 ° k ( ¿) cos ( θ+360 ° k ) +i sin¿ θ∈degrees z=r ¿
lii. i.
k =0,± 1, ±2, … ; θ∈radians
Multiplication of Polar Form liii. If z 1=r 1 ( cos θ1 +isin θ2 ) =r 1 ∠ θ 1
;
z 2=r 2 ( cos θ2 +isin θ 2) =r 2 ∠θ 2 liv. ii.
z 1 z 2=r 1 r 2 [ cos ( θ1 +θ2 ) +i sin ( θ1+ θ2 ) ]∨z 1 z2 =r 1 r 2 ∠ ( θ1 +θ2 )
Division of Polar Form lv. If z 1=r 1 ( cos θ1 +isin θ2 ) =r 1 ∠ θ 1
;
z 2=r 2 ( cos θ2 +isin θ 2) =r 2 ∠θ 2 lvi.
z1 r1 z r = [ cos ( θ 1−θ2 ) +i sin ( θ1−θ2 ) ]∨ 1 = 1 ∠ ( θ1−θ 2) z2 r2 z2 r2
7. Exponential Form of a Complex Number. z=r e i(θ+2 kπ) −exponential form , θ∈radians lvii. lviii. i.
General Exponential Form:
Multiplication of Exponential Form iθ iθ z 2=r 2 e lix. If z 1=r 1 e ; 1
z=r e 2
i(θ+2 kπ )
, k=0,± 1,± 2, ±3 …
i θ 1 +θ2)
z 1 z 2=r 1 r 2 e (
lx. ii.
Division of Exponential Form iθ z 2=r 2 ei θ lxi. If z 1=r 1 e ; 1
2
z 1 r 1 i( θ −θ ) = e z2 r2 1
lxii.
2
8. Powers and Roots of Complex Numbers (De Moivre’s Theorem) n n n n z =[ r ( cos θ+i sinθ ) ] =r ( cos nθ+i sin nθ ) =r ∠ nθ lxiii. 1 n
1 n
1 n
[ (
θ+2 πk θ+2 πk +i sin √ z=z =[ r ( cos θ+isin θ ) ] =r cos n
lxiv.
n
)
(
n
)]
lxv.Where: k =0,1, 2, 3 …(n−1) 9. Dot and Cross Product lxvi. Let z 1=x 1+i y 1 and lxvii.
Dot Product:
z 2=x 2+i y 2 be two complex numbers.
z 1 ∘ z 2=| z1||z 2|cos θ=x1 x 2+ y 1 y 2 lxviii.
1 ¿ ℜ ( z´1 z 2 )= ( z´1 z 2+z 1 z´2 ) 2
Cross Product: z 1 x z 2=| z1||z 2|cos θ=x 1 x2 − y 1 y 2
lxix.
lxx. lxxi.
¿ ℑ ( z´1 z 2 )=
1 ( z´ z −z z´ ) 2i 1 2 1 2
z 1 and z 2 are non-zero, then: and z 2 are perpendicular if z 1 ∘ z 2=0 and z 2 are parallel if z 1 x z 2=0 If
i.
z1
ii.
z1
iii. The magnitude of the projection of
z 1 and
iv. The area of parallelogram having sides 10. Exponential and Trigonometric Functions e z =e x+iy =e x (cos x+i sin y) i. ii.
e z =e z ln a
iii.
sin z=
eiz −e−iz 2i
iv.
cos z=
e iz + e−iz 2
z 2 is
z 1 and
|z 1 ∘ z2|/|z 2|
z 2 is
|z 1 x z 2|
v.
sin z=sin ( x+ iy )=sin x cosh y +i cos x sinh y
vi.
cos z=cos ( x +iy )=cos x cosh y −isin x sinh y
11. Hyperbolic Functions z −z e −e sinh z= i. 2 e z + e−z 2
ii.
cos z=
iii.
sinh z=sinh ( x+ iy )=sinh x cos y +icosh x sin y
iv.
cosh z =cosh ( x +iy ) =cosh x cos y−i sinh x sin y ln z=ln( x+iy)=ln r +i(θ+2 πk )
12. Logarithmic Functions: 13. Inverse Trigonometric Functions iz (¿ ± √1−z 2) i. sin−1 z =iln ¿
14. Inverse Hyperbolic Functions z ± √ z 2+ 1 (¿) i. −1 sinh z =ln ¿
ii.
iz ± √ z 2−1 ( ¿) −1 cos z=i ln ¿
iii.
tan −1 z=
ii.
z ± √ z 2−1 ( ¿) −1 cosh z=ln ¿
iii.
tanh −1 z=
−i 1+ iz ln 2 1−iz
(
)
−1 1+ z ln 2 1−z
( )
iv. c. Infinite Series – a series in which the number of terns is unlimited. It is denoted by the symbol u1 +u2 +u3 +… un +… or expressed by Σ−notation ∞
v.
u1 +u2 +u3 +… un +…= ∑ u n
1. Sum of Infinite Series:
n →1
S n=u 1+u 2+u 3+ … ; S=lim S n n→∞
2. Convergent Series. If the series has a sum S, if S n approaches a limit when n→∞ . 3. Divergent Series. If the limit does not exist. 4. Test for Convergence or Divergence of a Series i. Ratio Test
|u n+1| <1, the series converges . n →∞ |un|
If lim
If lim
|u n+1| |u n+ 1| >1∨if increases withou bound n →∞ |un| |un| vi.
the series converges .
|u n+1| =1, the test fails . n →∞ |un|
If lim
ii. Root Test n If lim √|u n|<1, the series converges . n →∞
If lim √n |u n|>1∨if lim √n |un|=∞ the series diverges .
If lim √|u n|=1, the test fails .
n →∞
n→∞
n
n →∞
vii. viii. ix. x. xi. xii. xiii. D. Plane and Spherical Trigonometry a. Trigonometry: tri = three, gonia = angle and metron = measurement. b. Kinds of Angels: Let A be ∠ A . i. ii. iii. iv.
Zero Angle: exactly 0° v. Straight Angle: A=180° Acute Angle: 0° < A < 90° vi. Reflex Angle: 180° < A < 360° Right Angle: A = 90° vii. One Revolution: A = 360° Obtuse Angle: 90° < A < 180° Complementary Angles are angles whose sum is 90°. Supplementary Angles are angles whose sum is 180°. Explementary Angles are angles whose sum is 360°. c. Units of Angles π 90 °= radians=100 grades=1600 mils 2 1 radian is the angle subtended by an arc of a circle whose length is one radius. d. Kinds of Triangle i. According to Angle Acute Triangle – all angles are acute angles Right Triangle – one angle is a right angle Obtuse Triangle – one of its angle is obtuse angle c A
B a C
b
ii.
According to Sides Isosceles Triangle – two sides are equal Scalene Triangle – no sides are equal Equilateral Triangle – all sides are equal e. Functions of a Right Triangle opp . side a i. sin θ= hypotenuse = c
iv . sec θ=
adj. side b = hypotenuse c
v . cscθ=
opp . side a = adj . side b
vi . cot θ=
ii .cos θ=
iii . tan θ=
hypotenuse c = opp . side a
hypotenuse c = adj. side b adj. side b = opp . side a
f. Pythagorean Theorem – “In a right triangle, the sum of the square of the sides is c 2=a 2+ b2 equal to the square of its longest side (hypotenuse)” g. Trigonometric Identities: h. Identity is a type of equation which is satisfied with any value of the variable/s. i. Conditional Equation – an equation that is satisfied by some value of variable/s. i. Basic Identity j. l. a a/c sin θ c c /c 1 • tanθ= = = • secθ= = = b b/c cos θ a a/c sinθ k.
m.
b b/c cos θ •cot θ= = = a a/c sin θ
c c /c 1 •csc θ= = = b b /c cos θ
ii. iii. iv. Pythagorean Relations 2 2 •sin θ+cos θ=1 v. vi.
ii .cos ( x ± y )=cos x cos y ∓ sin x sin y
x.
o. Double Angle Formula i. sin 2 x=2sin x cos x ii.
2
2
•1+cot θ=csc θ
• tan 2 θ+1=sec 2 θ
n. Sum and Difference of Two Angles viii. i. sin ( x ± y )=sin x cos y ± sin y cos x ix.
vii.
2
tan x ± tan y 1 ∓ tan x tan y
iv. 2 ¿ 2 cos x −1
2
cos 2 x=cos x−sin x iii. ¿ 1−2 sin 2 x
iii. tan ( x ± y )=
v.
iii . tan 2 x= vi.
2 tan x 1−tan 2 x
p. Half Angle Formula: Let 2 x =θ ,then x=θ/2
√ √
i.
θ 1−cos θ i. sin = 2 2
ii.
θ 1+cos θ ii .cos = 2 2
q. Powers of Functions 1−cos 2 x i. sin2 x= r. 2 ii .cos 2 x =
s.
√
θ 1−cos θ iii . tan = 2 1+ cos θ
iii.
iii . tan 2 x=
t.
1−cos 2 x 1+cos 2 x
1+cos 2 x 2
u. Product of Functions 1 i. sin x cos y = 2 [ sin ( x + y ) +sin ( x− y ) ] ii.
sin x sin y=
1 [ cos ( x − y )−cos ( x + y ) ] 2
1 iii. cos x cos y= 2 [ cos ( x + y ) +cos ( x− y ) ] v. Sum and Difference of Functions (Factoring Formulas) x+ y x− y w. i. sin x+ sin y=2 sin 2 cos 2 x. y. z.
( ) ( ) x+ y x− y ii .sin x−sin y =2 cos ( sin 2 ) ( 2 ) x+ y x− y iii . cos x +cos y=2cos ( cos ( ) 2 2 ) x+ y x− y iv . cos x−cos y =−2 sin ( sin ( ) 2 2 )
aa. v . tan x+ tan y=
ab. sin( x+ y) cos x cos y
ac. Oblique Triangle – is any triangle that is not a right triangle. ad. Consider ∆ ABC :
vi . tan x−tan y =
c A
b
ae. B af. ag. a ah. ai. C
sin(x− y ) cos x cos y
i. Sine Law: In any triangle, the ratio of any side to the sine of its opposite angle is
constant. This constant ratio is the diameter of the circle circumscribing the triangle. a b c = = aj. sin A sin B sinC i. Cosine Law: In any triangle, the square of any side is equal to the sum of the square of the two other sides minus twice their product to the cosine of its included angle. ak. am. 2 2 2 2 2 2 a =b +c −2 bc cos A c =a + b −2 ab cos C al. b2=a2 +c 2−2 ac cos B i. Law of Tangents ( A−B ) /2 [¿] tan tan [( A + B)/2] ii. a−b =¿ a+b
iii.
iv.
( C− A ) /2 [¿] tan tan [(C+ A)/2] c−a =¿ c +a
( B−C ) /2 [¿] tan tan [(B+ C)/2] b−c =¿ b+c
v. vi. Mollweide’s Equations vii. a−b sin [( A−B)/ 2] = c cos (C/2)
viii. a+b cos [( A−B)/ 2] = c sin(C /2)
an. Spherical Triangle – a triangle enclosed by arcs of three great circles of a sphere. The sum of the interior angles of a spherical triangle is greater than 180° but less than 180 °< ( A +B+ C ) <540 ° 540°. ao. Area of Spherical Triangle: 2 πR E A= i. 180 ° ii.
Where : E isthe spherical excess∈degrees
iii.
E= A +B+ C−180°
iv.
tan
E s s−a s−b s−c = tan tan tan tan 4 2 2 2 2
v.
s=
a+ b+c 2
√
vi. For an arc of a great circle of the earth, the distance equivalent to 1 minute (0°1’) of the arc is one (1) nautical mile (6080 ft.). ap. Right Spherical Triangle c
i.
