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Chapter 10 to 18 Notes
Chapter 10
Distance formula- The distance between any two points (x1,y1) and (x2,y2) is given by d= √(x1-x2)2+(y1-y2)2
Conic section- The nonempty intersection of any plane with a cone.
Unit circle- A circle with a radius of 1.
Ellipses- The set of all point P in a plane such that the sum of the distances from P to two fixed points f 1
and f2 is constant. Each fixed point is called a focus.
Major axis- The larger axis in an Ellipse
Minor axis- The smaller axis in an Ellipse
Center- The set of all points in a plane that are at constant distance from a fixed point in that plane. The
fixed point is the center of the circle.
Vertices- Intersections of the major and minor axes in an Ellipse. They are points a and –a in a Hyperbola.
Branches- Each singular “parabola” in a Hyperbola
Transverse axis- The line segment joining points a and –a.
Conjugate axis- The line segment joining b and –b.
Asymptotes-A line in an asymptote to a curve if the curve gets very close to the line as the distance from
the center increases.
Eccentricity- The ratio of the distance of any point on a conic section from a focus to its distance from the
corresponding directrix. This ratio is constant for any particular conic section.
Conjugate- Complex pairs such as 1+2i and 1-2i.
Caternary- A curve formed by something such as a cable on a suspension bridge that only supports its own
weight.
Simplify the Problem- A problem solving strategy that helps evade the use of large numbers making it
extremely difficult to come up with a direct answer.
Distance- The amount of units separating two points.
Midpoint- The center point between two points that would be the endpoints of a segment.
Hyperbola-The set of all point P in a plane such that the absolute value of the difference of the distances
from P to two fixed points F1 and F2 is constant. The fixed points F1 and F2 are the foci.
Parabola- The set of all points equidistant from a fixed point (the focus) and a fixed line in a plane. The
graph of the quadratic equation ax2+bx+c=0, a≠0, is a parabola.
Distance Formula
The distance between two points (x1,y1) and (x2,y2) is given by
d= √(x1-x2)2+(y1-y2)2
Midpoint formula
If the coordinates of the endpoint of a segment are (x1,y1) and (x2,y2), the n the coordinates of the midpoint
are
x1+x2 y1+y2
2 ,
2
A circle is the set of all points in a plane that are at a constant distance from a fixed point in that plane. The
fixed point is the center of the circle.
The equation (in standard from) of the circle centered at the origin with radius r is x2+y2=r2.
The equation (in standard from) of the circle with center (h,k) and radius r is
(x-h)2+(y-k)2=r2.
The set of all point P in a plane such that the sum of the distances from P to two fixed points F 1 and F2 is
constant. Each fixed point is called a focus (plural: foci) of the ellipse. r 1 and r2 are called focal radii.
The equation, in standard form, of an ellipse centered at the origin with foci on an axis c units from the
origin is
x2 + y2
a2 b 2
=
1
(major axis horizontal)
x2 + y2
b 2 a2
=
1
(major axis vertical)
where c2=a2-b2
The equation, in standard form, of an ellipse centered at the origin with foci on an axis c units from the
origin is
(x-h)2 + (y-k)2
a2
b2
=
1 (major axis horizontal) (x-h)2 + (y-k)2
b2
a2
=
1
(major axis vertical)
where c2=a2-b2
A hyperbola is the set of all points P in a plane such that the absolute value of the difference of the
distances from P to two fixed points, f1 and f2, is constant. The fixed points F 1 and F2 are the foci. The
midpoint of the segment F1F2 is the center. r1 and r2 are the focal radii.
The equation, in standard form, of an Hyperbola centered at the origin with foci on an axis c units from the
origin is
x2 - y2
a2 b 2
=
1
(major axis horizontal)
x2 - y2
b 2 a2
=
1
(major axis vertical)
where c2=a2+b2
The equation, in standard form, of a Hyperbola centered at (h,k) with foci on an axis c units from (h,k) is
(x-h)2 - (y-k)2
a2
b2
=
1 (major axis horizontal) (x-h)2 - (y-k)2
b2
a2
=
1
(major axis vertical)
where c2=a2+b2
The equation of a hyperbola with asymptotes on the axes is xy=c, where c≠0.
A parabola is a set of all point P in a plane equidistant from a fixed line and a fixed point in the plane. The
fixed line is called the directrix and the fixed point is called the focus.
A parabola with focus at (0,p) and vertex at (0,0) has directric y=-p.
The equation (in standard form) of a parabola with focus at (0,p), directric y=-p, vertex (0,0), and y-axis as
the only line of symmetry is x2=4py.
The equation (in standard form) of a parabola with focus (p,0), directrix x=-p, vertex (0,9(, and x-0 axis as
the line of symmetry is y2=4px.
If a parabola is translated so that its vertex is (h,k) and its axis of symmetry is parallel to the y-axis, it has
an equation of (x-h) a2=4p(y-k), where the focus is (h, k+p) and the directrix is y=k-p.
If a parabola is translated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has
an equation of (y-k)2=4p(x-h), where the focus is (h+p, k) and the directrix is x=h-p.
Consider the second-degree equation Ax2+By2+Cx+Dy+E=0.
1. If A=B, the equation may define a circle.
2. If A and B has the same sign and A≠B, the equation may define an ellipse.
3. If A and B has different signs, the equation defines a hyperbola.
If either A or B is 0, the equation defines a parabola.
Chapter 11
Leading coefficient- The coefficient of the term of the highest degree in a polynomial.
Factor- A divisor is a factor when it is divided into the dividend and the remainder obtained is zero.
Descartes’ rule of signs- A method to find the amount of positive real roots of a polynomial.
Variations of sign- the number of times successive coefficients have difference signs in a polynomial
Cubic- A polynomial of the third degree
Quartic- Graphs of the fourth-degree
Work Backward- A problem solving strategy using information already known to go in reverse order to
come up with the needed information.
Root of a polynomial- Any number that makes the polynomial zero.
Zero of a polynomial- A number that when substituted into a polynomial function makes the function zero.
Remainder Theorem- For a polynomial P(x), the functional value P(r) is the remainder when P(x) is divided
by x-r.
Remainder- The resulting amount in a quotient that is a fractional part over the divisor and is added onto
the quotient.
Factor Theorem- For a polynomial P(x), if P(r)=0, then the polynomial x-r is a factor of P(x).
Multiplicity- The amount of times a number is a root in a polynomial.
Complex roots- Numbers that have an i part that occur in conjugate pairs in polynomials with real
coefficients.
Irrational roots- An number root that occurs in pairs in a polynomial with rational coefficients.
Rational roots- rational numbers that are roots of a polynomial
Positive real roots- Real number roots of which amount in a polynomial with real coefficients in either the
same as the number of variations of signs of P(x) or less than the number of variations by a positive even
integer.
Negative real roots- Real number roots of negative value that occur in a polynomial of real coefficients in
either the number of variation of the sign of P(-x) or less than the number of variations of sign by a positive
even integer.
If a number n is a solution of a polynomial equation P(x)=0, then n is called a root of the equation
If a number n, when substitutes for x, makes a polynomial function P(x) zero, then n is called a zero of the
function.
The Remainder Theorem
For a polynomial P(x), the function value P® is the remainder when P(x) is divided by x-r.
The Factor Theorem
For a polynomial P(x), if P(r)=0, then the polynomial x-r is a factor of P(x).
The fundamental Theorem of Algebra
Every polynomial of degree n>1, with complex coefficients, has at least one linear factor.
Every polynomial of degree n>1, with complex coefficients, can be factored into exactly n linear factors.
If a polynomial P(x) of degree greater than or equal to 1 with real coefficients has a complex number a+bi
as a root, then its conjugate a-bi is also a root. (complex roots occur in conjugate pairs.)
Suppose P(x) is a polynomial with rational coefficients of degree greater than or equal to 1. Then if either
of the following is a root, so is the other: a+c√b, a-c√b, where a, b, and c are rational number and √b is
irrational.
Rational Roots Theorem
Let P(x)=anxn+an-1xn-1+…+a1x+a0 where all coefficients are integers. Consider a rational number denoted
by c/d, where c and d are relatively prime. For c/d to be a root of P(x), c must be a factor of a 0 (the
constant) and d must be a factor of an (the leading coefficient).
Finding Rational Roots
To find possible rational roots of P(x)=anxn+an-1xn-1+… +a0
1. find all factors of an and a0.
2. find each rational number c/d such that d is a factor of an and c is a factor of a 0.
3. if an is 1, test only the factors of a0.
4. test each c/d using synthetic division.
Descartes’ Rule of Signs
The number of positive real roots of a polynomial P(x) with real coefficients is
a. The same as the number of variations of sign of P(x), or
b. Less than the number of variations of sign of P(x) by a positive even integer
Corollary to Descartes’ Rule of Signs
The number of negative real roots of a polynomial P(x) with real coefficients is
a. the number of variations of sign of P(-x), or
b. less than the number of variations of sign P(-x) by a positive even integer.
Graphs of Higher-Degree Polynomial Functions
1. Every polynomial function has its domain the set of real numbers.
2. The graph of any function is a continuous unbroken curve that must pass the vertical line
test.
3. Unless a polynomial function is linear no part of its graph is straight.
4. A polynomial of degree n cannot have more than n real roots. This means that the graph
cannot cross the x-axis more than n times.
Graphing Polynomials
1. Look at the degree of the polynomial and its leading coefficient. This gives a lot of information
about the general shape of the graph.
2. Look for symmetries, as covered in Chapter 9. When symmetrical points occur, the rest of the
graph can be plotted quickly.
3. Make a table of values using synthetic division.
4. Find the y-intercept and as many x-intercepts as possible (the latter are roots of the polynomial).
In doing this, recall the theorems about roots, including Descartes’ rule of signs.
5. Plot the points and connect them appropriately.
Chapter 12
1. inverse - exchanging first and second coordinates of each ordered pair
2. exponential function - the function f(x) = ax, where a is some positive real-number constant different
from 1
3. logarithmic function - inverse of an exponential function
4. common logarithms - these are Base 10 logarithms
5. log - abbreviation for logarithmic function base 10.
6. characteristic - integer part of logarithm, exponent in the scientific notation
7. mantissa - other part of log, number between 0 and 1
8. antilogarithm - reverse process of log
9. interpolation - estimating values between those listed in common log table
10. linear interpolation - simplest and most common interpolation
11. exponential equation - an equation with variables in exponents
12. logarithmic equation - equations that contain logarithmic expressions
13. natural logarithm - logs to the base e
Interchanging x and y in the equation of a relation produces an equation of the inverse relation.
For any function f whose inverse is a function, f^-1(f(a)) = a for any a in the domain of f. Also f(f^-1))=a
for any a in the domain of f^-1.
For any positive number a, where a does not equal 1, alogax = a, for any positive number x and logaax = a, for
any number x.
For any positive numbers x and y, logaxy = logax + logay where a is any positive number different from 1.
For any positive number x, any number p, and any logarithm base a, logaxp = p*logax
For any positive numbers x, y, and any logarithm base a, log a (x/y) = loga x - loga y
For any bases a and b, and any positive number M, log b M = loga M/loga b
In M = log M/log e and log M = In M/In 10
Chapter 14
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Infinite sequence- a sequence, ordered set of numbers, that does not end
Term- a number in a sequence
n-th term or general term- the term which will allow the deduction of other terms
recursion- a sequence in which each subsequence term is related to the term before it in the sequence
partial sum- sum of a part of an infinite sequence called a series
infinite series- the sum of a sequence of numbers
arithmetic mean- aka average
arithmetic means- averages between values to make an arithmetic sequence
arithmetic series- a series associated with an arithmetic sequence in which the sum of the terms is
found
harmonic sequence- a sequence whose reciprocals form an arithmetic sequence
harmonic mean- a value which creates a sequence whose reciprocals form an arithmetic sequence.
geometric mean- product values between two values to make a geometric sequence
limit- the value (1) to which a infinite geometric sequence approaches if the common ratio is less than
one
principle of mathematical induction- the logic in which to obtain whether or not a sequence is true
14-1
A sequence is an ordered set of numbers
An indicated sum of the terms of a sequence.
Sn=a1+a2+a3+…+an
Is called a series
14-2
A seqeuce in which a constant d can be added to each term to get the next term is called an arithmetic
sequence. The constant d is called the common difference.
The n-th term of an arithmetic sequence is given by
an=a1+(n-1)d
The sum of the first n terms of an arithmetic series is given by
Sn=n/2 (a1+an)
The sum of the first n terms of an arithmetic series is given by
Sn= n/2 [2a1+(n-1)d]
14-3
A sequence in which a constant r can be multiplied by each term to get the next is called a geometric
sequence
The constant r is called the common ratio
In a geometric sequence, the n-th term is given by an=a1rn-1
The sum of the first n terms of a geometric series is given by
Sn= a1(1-rn) / (1-r)
14-4
If, in an infinite series, Sn approaches some limit as n becomes very large, that limit is defined to be the
sum of the series. If an infinite series has a sum, it is said to converge or to be convergent.
An infinite geometric series is convergent and thus has a sum if and only if |r|<1 (the absolute value of the
common ratio is less than 1).
The sum of an infinite geometric series, with |r|<1, is given by S=a1/ (1-r)
14-5
The principle of Mathematical Induction
If Pn is a statement concerning the positive integers n and
A. P1 is true,
B. Assuming Pk is true implies that pk+1 is true
Then Pn must be true for all positive integers m.
Chapter 15
1. combinatorics - study of ways numbers can be arranged
2. tree diagram - method of listing possibilities
3. outcome - result of situations
4. event - subset of outcomes
5. compound event - several events together
6. permutation - set of n objects is an ordered arrangement of the objects
7. n factorial or n! - n(n-1) x … x 3 x 2 x 1
8. combinations - only with the number of ways we can select elements from a set
9. sample space - set of all possible outcomes
10. probability - given be P(E) where P(E) = m/n
11. intersection - set of elements that contains those elements common to both A and B
12. union - set of elements that belong to either A or B
13. mutually exclusive - two events which have no common elements
14. compound probability - probably of any compound events occuring
15. independent - A having no effect on B and vice versa
16. theoretical probability - actual event
17. experiment - design an event
18. experimental probability - event determined by experiment
19. simulate - creating with a model
20. Monte Carlo - methods that use simulations to determine probability
The Fundamental Counting Principle
In a compound event in which the first event may occur, n1 different ways, the second event may occur in
n2 different ways and so on, and the k-th event may occur in nk different ways, the total number of ways the
compound event may occur is n1*n2*n3 ..*nk
The total number of permutations of a set of n objects is given by nPn = n*(n-1)*(n-2)…*3*2*1
For any natural number n, n!=n(n-1)!
The number of permutations of a set of n objects taken r at a time, denoted nPr, is given by nPr = n!/(n-r)!
The number of orderings of n objects taken r at a time, with repetition, is n r.
The number of permutations, P, of n objects taken n at a time, with r objects alike, s of another kind alike, t
of another kind alike, is P = n!/(r!s!t!)
The number of circular permutations of n objects is n!/n, or (n-1)
The number of combinations of a set of n objects taken r at a time is (n choose r) = n!/r!(n-r)!
The Binomial Theorem
For any binomial (a+b) and any whole number n, (a + b) n = (n choose 0)an + (n choose 1)an-1(b) + (n choose
2)an-2(b2) ……….
The total number of subsets of a set with n members is 2 n.
The probability of any event is a number from 0 to 1. If an event cannot occur its probability is 0. If an
event is certain to occur its probability is 1.
If A and B are events from a sample space S, then P(A U B) = P(A) + P(B) - P(A B)
Two events A and B are independent if and only if P(A B) = P(A) x P(B)
Chapter 16
1.
stem-and-leaf diagram- A chart used to organize data.
2.
ranked stem-and-leaf diagram- A stem-and-leaf diagram organizing the data from least to greatest.
3.
frequency distribution- A table showing the frequency, or number of occurrences, of a particular
set of data.
4.
relative frequency- The frequency divided by the total number of values.
5.
mean- The average of a set of values.
6.
average- Also known as the mean.
7.
box and whisker plot- A chart used to display a set of data.
8.
upper hinge- The median of the upper half of the data in a box and whisker plot.
9.
lower hinge-The median of the lower half of the data in a box and whisker plot.
10.
bimodal- An instance where a set of data has two different modes.
11.
trimmed- A mean that is found by ordering the values from least to greatest, deleting an equal
number from each end and finding the mean of the remaining values.
12.
trimming percentage- The percentage of deleted values from a trimmed mean.
13.
mean deviation- The measure of spread or variation; the average amount that a set of data deviates
from the mean.
14.
variance- A measure of the amount of variation in a set of data. It is found by taking the mean of
all the squared distances of the values from the mean.
15.
σ2- Also known as variance, the square of standard deviation.
16.
standard deviation- The square root of the variance.
17.
σ- Symbol for standard deviation.
18.
normal distribution- Describes the relative arrangement of a set of elements such that they fit the
normal curve.
19.
normal curve- A symmetric bell shaped curve describing the normal distribution.
20.
standard normal distribution21.
sampling- Effectively drawing and examining samples of data.
22.
population- A group of data collected.
23.
representative- Where a sample would be satisfying and equal probability of all people included.
24.
biased- A data overly influenced by factors that are related to the survey.
25.
strata- A distinct subgroup that is drawn from a representative sample.
26.
stratified random sampling- It is used to assure that the sample has the same characteristics as the
population.
27.
significant- Where something has a certain level of error.
28.
level of significance- The percentage that states whether chance played a part in the results rather
than being directly affected by something else.
29.
chi-square (χ2)-A hypothesis test used to determine whether a given set of data differs from what
one would expect by chance.
30.
Null hypothesis- There is no statistical difference between the expected and observed data. Thus,
the observed results occurred by chance.
A random sample is a sample selected such that
1. each object (person) in the population has an equal chance of being selected for the sample
2. each object (person) in the sample is chosen independently of any other objects in the sample
Chapter 17
1. Trigonometric functions- functions which deal with the measures of the sides of a right triangle
2.
Trigonometry- “triangle measurement”
3.
Hypotenuse- side opposite the right angle in a right triangle
4.
Opposite- leg across from the unknown angle
5.
θ (theta)- symbol for unknown angle measurement
6.
Adjacent- leg which is not across from the unknown angle
7.
Terminal side- rotating ray in a circular path of a triangle side on a coordinate plane
8.
Initial side- the positive half of the x-axis
9.
Radian- 1 radian is distance around the circle from the initial side to the terminal side
10.
Unit circle- a circle with radius of 1
11.
Angular speed- the amount of rotation per unit of time
12.
Periodic- functions with repeating patterns
13.
Period- the distance from one point to a point in which will repeat a function in a period function
14.
Amplitude- half the difference between its maximum and minimum function values which is
always positive
15.
Asymptotes- a line is an asymptote to a curve if the curve gets very close to the line as the distance
from the center increases
16.
Identities-an equation that is true for all acceptable replacement for the variables.
17.
Cofunctions- functions which are inverses of each other
17-1
sine function sin θ= length of the side opposite θ
length of the hypotenuse
cosine function cos θ= length of the side adjacent θ
length of the hypotenuse
tangent function tan θ= length of the side opposite θ
length of the side adjacent θ
cotangent function cot θ=
1
= length of the side adjacent θ
tan θ length of the side opposite θ
secant function sec θ= 1 =
cos θ
length of the hypotenuse
length of the side adjacent θ
cosecant function csc θ= 1 =
sin θ
length of the hypotenuse
length of the side opposite θ
17-2
The reference angle for rotation is the acute angle formed by the terminal side and the x-axis
17-3
The radian measure θ of a rotation is the ratio of the distances s traveled by a point at a radius r from the
center of rotation to the length of the radius
Cofunction identities
sin θ= cos (90- θ)
cos θ= sin (90- θ)
csc θ= sec (90- θ)
sec θ= csc (90- θ)
tan θ= cot (90- θ)
cot θ= tan (90- θ)
The linear speed v of a point of distance of r from the center of rotation is given by
v=rω
Where ω is the angular speed in radians per unit of time
17-5
If a function f has the property that f(x+p)=f(x) for all x in the domain where p is a constant, then f is said to
be periodic. The smallest positive number p, (if there is one) for which f(x+p)=f(x) for all x, is the period
of the function.
The sine function is periodic, with period 2π
sin θ = sin (θ+2 π) = sin (θ-2 π) = sin (θ+4 π), and so on, or sin θ= sin (θ±2 πk) where k is an integer.
cos θ = cos (θ+2 π) = cos (θ-2 π) = cos (θ+4 π), and so on, or cos θ = cos (θ±2 πk) where k is an integer.
17-6
The Quotient Identities
tan θ = sin θ / cos θ, cos θ ≠ 0
cot θ = cos θ / sin θ, sin θ ≠ 0
The Pythagorean Identities
sin² θ + cos² θ ≡ 1
1 + cot² θ ≡ csc² θ
1+ tan² θ ≡ sec² θ
The Cofunction Identities
sin (θ + π/2) ≡ cos θ
sin (θ - π/2) ≡ -cos θ
sin (π/2 - θ) ≡ cos θ
cos (θ + π/2) ≡ -sin θ
cos (θ - π/2) ≡ sin θ
cos (π/2 - θ) ≡ sin θ
Chapter 18
1. double-angle identities - identities involving sin 2θ or cos 2θ
2. principal values - values in restricted ranges
3. trigonometric equation - when an equation contains a trigonometric expression with a variable such as
sin x
4. angle of elevation - angle between horizontal and a line of sight
5. angle of depression - angle between horizontal and a line of sight
6. law of sines - a property used in order to solve oblique triangles
7. law of cosines - second property used in order to solve oblique triangles
8. absolute value of a complex number - distance r from the origin a + bi
9. argument- angle ø
10. rectangular notation - another form of a trigonometric notation
11. Trigonometric/polar notation - r (cos θ + i sin θ)
18-1
Cos (α-β ) = cos α cos β + sin α sin β
Cos (α+β) = cos α cos β - sin α sin β
Sin (α-β) = sin α cos β - cos α sin β
Sin (α+β) = sin α cos β + cos α sin β
Tan (α-β) = (tan α - tan β)/(1+tan α tan β)
Tan (α+β) = (tan α + tan β)/(1-tan α tan β)
18-2
Sin 2θ = 2 sin θ cos θ
Cos 2θ = cos² θ - sin² θ
Cos 2θ = 1-2sin² θ - 1
Cos 2θ = 2 cos² θ - 1
Tan 2θ = (2 tan θ)/(1-tan² θ)
Sin² θ = (1-cos 2θ)/(2)
Cos² θ = (1+cos 2θ)/(2)
Tan² θ = (1-cos 2θ)/(1 + cos 2θ)
Φ=2θ
sin Φ/2 = ± √(1 - cos Φ)/2)
cos Φ/2 = ± √ (1 + cos Φ)/2)
tan Φ/2 = ± √((1- cos Φ)/(1 + cos Φ))
Tan Φ/2 = sin Φ/1 + cos Φ
Tan Φ/2 = 1 - cos Φ/ sin Φ
The Law of Sines
In any triangle ABC, a/sin A = b/sin B = c/sin C (The sides are proportional to the sines of the opposite
angles.)
The Law of Cosines
In any triangle ABC,
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B
c² = a² + b² - 2ab cos C
(In any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the
product of those sides and the cosine of the included angle.)
For any complex numbers r1 cis θ1, and r2 cis θ2….
(r1 cis θ1)(r2 cisθ2) = r1*r2 cis (θ1 + θ2)
For any complex numbers r1 cis θ1 and r2 cis θ 2…
r1cisθ1 / r2cis θ2 = r1/r2 cis (θ1 - θ2)
DeMoivre's Theorem
For any complex number r cis θ and any natural number n, (r cis θ) n = rn cis nθ
The n-th roots of a complex number r cis θ are given by:
r1/n cis(θ/n + k360/n), where k= 0 ,1, 2 .., n-1
Trigonometric Identities
-sin x ≡ sin (-x)
cox x ≡ cos (-x)
-tan x ≡ tan (-x)
sin ( x + π/2) ≡ cos x
sin ( x - π/2) ≡ -cos x
cos ( x + π/2) ≡ -sin x
cos ( x - π/2) ≡ sin x
sin 2x ≡ 2 sin x cos x
cos 2 x ≡ cos x–sin 2 x ≡ 1–2 sin2 x ≡ 2cos2 x–1
tan 2x = 2tan x / 1-tan2 x
sin2 x ≡ 1-cos 2x / 2
cos2 x ≡ 1+cos 2x / 2
tan2 x ≡ 1–cos 2x / 1+cos 2x
2
sin2 x+ cos2 x ≡ 1
1 + tan2 x ≡ sec2 x
1 + cot2 x ≡ csc2 x
sin (α + β ) ≡ sin α cos β + cos α sin β
sin (α - β ) ≡ sin α cos β - cos α sin β
cos (α + β ) ≡ cos α cos β - sin α sin β
cos (α - β ) ≡ cos α cos β + sin α sin β
tan (α + β ) ≡ tan α + tan β / 1 - tan α tan β
tan (α – β ) ≡ tan α - tan β / 1 + tan α tan β
sin x/2 ≡ ±√( 1–cos x / 2)
cos x/2 ≡ ±√( 1+cos x / 2)
tan x/2 ≡ ±√( 1–cos x / 1+cos x) ≡ sin x / 1+cos x ≡ 1-cos x / sin x