# How do I distinguish sin mod x

## Math 9 - Real Numbers (LOERn)

### Basic mathematical terms

A small dictionary of basic mathematical terms.

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 An image is called an axis reflection on a symmetry axis, ifevery point on the axis is a fixed point in the figure andevery point outside the axis forms a line with its image point, which is perpendicular to the axis and is bisected by it. Method for solving a system of equations.Two sorted equations are cleverly added, so that one of the variables is omitted in the sum.Example:I: \ (\ small 4x + 2y = 3 \)II: \ (\ small 2x-2y = 9 \)I + II: \ (\ small 6x + 0 = 12 \)from which follows: \ (\ small x = 2 \)Then this partial result has to be inserted into one of the two output equations and this has to be solved according to the second quantity. Two triangles are similar to each otherwhen they match in all of their angles.One writes: \ (\ small \ triangle ABC \ sim \ triangle DEF \)It is sufficient if:The triangles match at two angles orthe triangles coincide in a distance ratio and an angle orthe triangles coincide in two route ratios.Congruence is a special case of similarity.
 Short side in a right triangle that forms a vertex to the angle under consideration.
 The arccosine is the inverse function of the cosine. The arccosine is used to assign the corresponding angle to a ratio of the distance between the adjacent side and the hypotenuse. \ (\ arccos \ frac {b} {c} = \ cos ^ {- 1} \ frac {b} {c} = \ alpha \)\ (\ arccos \) does not expect an angle, but a number as an argument.\ (\ cos ^ {- 1} x \) is just another notation for \ (\ arccos x \) and does not correspond to the reciprocal of \ (\ cos \ varphi \)!
 The arcsine is the inverse function of the sine. The arc sine is used to assign the corresponding angle to a distance ratio of the opposite cathetus and hypotenuse. \ (\ arcsin \ frac {a} {c} = \ sin ^ {- 1} \ frac {a} {c} = \ alpha \)\ (\ arcsin \) does not expect an angle, but a number as an argument.\ (\ sin ^ {- 1} x \) is just another notation for \ (\ arcsin x \) and does not correspond to the reciprocal of \ (\ sin \ varphi \)!
 The arctangle is the inverse function of the tangent. The arctangent thus assigns the associated angle to a distance ratio of the opposite side and the adjacent side. \ (\ arctan \ frac {a} {b} = \ tan ^ {- 1} \ frac {a} {b} = \ alpha \) \ (\ arctan \) does not expect an angle, but a number as an argument. \ (\ tan ^ {- 1} x \) is just another notation for \ (\ arctan x \) and does not correspond to the reciprocal of \ (\ tan \ varphi \)!
 "To associate" means "to link" or "to connect".In the associative law, two parts of a term are more closely linked than with the rest.The associative law applies toaddition\ (a + b + c = (a + b) + c = a + (b + c) \),multiplication\ (a \ cdot b \ cdot c = (a \ cdot b) \ cdot c = a \ cdot (b \ cdot c) \),but not for division and difference.
 If a term in brackets is multiplied by a factor, each summand in the brackets must be multiplied individually by the factor.Example:\ (a \ cdot (b + c) = ab + ac \)Numerical example:\ (3 \ times (2 + 4) = 6 + 12 = 18 = 3 \ times 6 \)
 A directed graph with exactly one root element.The elements of the tree diagram are referred to as nodes or leaves, the connections between the elements as edges or branches.A tree diagram is used to illustrate multi-level processes or multi-level random experiments (or data structures in computer science).
 The binomial formulas give possibilities to convert products of sums and differences into sums or differences.3 binomial formulas are part of basic knowledge:\ (\ small (a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2 \)\ (\ small (a-b) ^ 2 = a ^ 2 -2ab + b ^ 2 \)\ (\ small (a + b) (a-b) = a ^ 2 - b ^ 2 \)However, these are only the special cases for 2 factors.The binomial theorem shows what to do with more than 2 factors.
 The following applies:\ ((a + b) ^ n = \ sum_ {k = 0} ^ n {n \ choose k} a ^ {n-k} b ^ k \)written out:\ (\ small (a + b) ^ n = {n \ choose 0} a ^ n + {n \ choose 1} a ^ {n-1} b ^ 1 + \ ldots + {n \ choose n-1} a ^ 1 b ^ {n-1} + {n \ choose n} b ^ n \)The symbols \ (\ small {n \ choose k} \) are the binomial coefficients, which can also be found in Pascal's triangle.Example:\ (\ small (2 + x) ^ 4 = 1 \ times 2 ^ 4 + 4 \ times 2 ^ 3 \ times x + 6 \ times 2 ^ 2 \ times x ^ 2 + 4 \ times 2 \ times x ^ 3 + 1 \ cdot x ^ 4 \)so: \ (\ small (2 + x) ^ 4 = 16 + 32x + 24x ^ 2 + 8x ^ 3 + x ^ 4 \).
 Two sizes \ (x \) and \ (y \) are directly proportional to each other if a doubling, tripling, ... of the size \ (x \) results in a doubling, tripling, ... of the size \ (y \)."The bigger \ (x \), the bigger \ (y \)."If there is direct proportionality, then there is equality of quotients:\ (\ frac {y} {x} = c \)\ (c \) is the constant of proportionality and the following applies: \ (y = c \ cdot x \).In the graphical representation of the indirect proportionality, there are straight lines through the origin.
 from lat. discernere: separate, distinguishdistinguishable, separable, countableA set is discrete if the number of its elements can be counted.Opposite to "continuous".
 The discriminant is the radicand in the formula for solving quadratic equations.\ (D = b ^ 2-4ac \)The discriminant shows how many solutions a quadratic equation has:\ (D> 0 \): two solutions\ (D = 0 \): a solution\ (D <0 \): no solution
 "distribuere" means "to distribute".The distributive law applies when a sum is a factor in a product.\ (a \ cdot (b + c) = ab + ac \)\ ((a + b) \ cdot c = ac + bc \)Every summand in brackets must be multiplied by the factor outside of brackets.The following also applies: \ ((a + b) \ div c = a \ div c + b \ div c \)But: \ (a \ div (b + c) \ ne a \ div b + a \ div c \)
 A dragon square is an axially symmetrical square, the diagonals of which are perpendicular to each other. Method for solving a system of equations.An equation is solved for a variable. Then in the second equation this variable is replaced by the complete term found in the first step. This gives you an equation that only depends on one variable.Example:I: \ (4x + y = 10 \)II: \ (3x + 5y = 16 \)I resolved for y: \ (y = 10-4x \)(Forward) insertion in II: \ (3x + 5 (10-4x) = 16 \)Simplify: \ (17x + 50 = 16 \)Solve for x: \ (x = 2 \)Insert backwards: \ (y = 10-4 \ cdot2 = 2 \)\ (\ mathcal {L} = \ lbrace (2; 2) \ rbrace \)
 An elementary event is an event that consists of only one result.Example: \ (A = \ lbrace \ omega \ rbrace \) is an elementary event.
 An event is a set of results from a random experiment and thus a subset of the result space.If \ (A \) is an event, then \ (A \ subset \ Omega \) applies.The safe event is \ (\ Omega \) itself.The impossible event is the empty set \ (\ lbrace \ rbrace \).
 An element of the result space \ (\ Omega \).Concrete result of a random experiment.Several results can be combined into one event.
 Every chance experiment involves a set of all possible outcomes. This set is called the result space \ (\ Omega \).\ (\ Omega \) canfinite \ (\ Omega = \ lbrace \ omega_1; \ omega_2; \ omega_3 \ rbrace \)or infinite \ (\ Omega = \ lbrace \ omega_1; \ omega_2; \ omega_3; \ ldots \ rbrace \)discrete \ (\ Omega = \ lbrace 1; 2; 3 \ rbrace \)or be continuous \ (\ Omega = [0; 1] \).
 If you just want to show it off in a crowd at least one There is an element that fulfills a statement, then it is completely sufficient a such Specify element.A single example is actually sufficient as evidence here.However, if you want to show that a statement of only exactly oneem element is fulfilled, that is not enough. Then you have to find that one element and show that the statement is correct. But then you have to show for all the others that they do not meet the statement.
 At the Factoring or Exclude you turn a sum into a product.This happens because common factors contained in the summands are "extracted" from the sum.Factoring is the opposite of multiplying.Example:\ (ab + ac = a \ cdot (b + c) \)Numerical example:\ (12 + 18 = 6 \ cdot (2 + 3) \)
 A recursively defined sequence of numbers named after Leonardo da Pisa.\(1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …\)This sequence has the remarkable property that every number is the sum of its two predecessors.The following applies to the \ (n \) th Fibonacci number:\ (f_0 = 1 \); \ (f_1 = 1 \);\ (f_n = f_ {n-1} + f_ {n-2} \) for \ (n \ geq 2 \).
 A function is one unambiguous (injective) Illustration from one crowd to another.Each element \ (x \) from the definition set \ (\ mathcal {D} \) is assigned exactly one \ (y \) in the value set \ (\ mathcal {W} \).How this assignment is made can be noted in three ways:as a set of value pairs \ (\ left \ lbrace \ left (x_1; f (x_1) \ right); \ left (x_2; f (x_2) \ right); \ ldots \ right \ rbrace \)as mapping rule \ (x \ mapsto y = \ ldots \)as a function equation \ (f (x) = y \)With \ (f (x) \) one denotes the function term, with \ (\ mathcal {G} _f \) the graph of the function graph.
 Side in the right triangle that is opposite to the angle under consideration.According to the naming convention, \ (a \) is the opposite side to \ (\ alpha \) and \ (b \) is the opposite side to \ (\ beta \).
 Method for solving a system of equations.Two equations are solved for the same term and then set equal. One variable is omitted.Example:I: \ (x + y = 2 \)II: \ (y = x -8 \)I resolved for \ (y \): \ (y = 2-x \)Equating: \ (2-x = x-8 \)Solve for \ (x \): \ (x = 5 \)Insert backwards: \ (y = 2-5 = -3 \)\ (\ mathcal {L} = \ lbrace (5; -3) \ rbrace \).
 Using several known equations, the aim is to find a combination of values ​​for several variables that solve all given equations simultaneously.In linear systems of equations, none of the variables occurs as a power. These are particularly easy to solve.Important solution methods are:The installation processThe equation procedureThe addition methodThe Gaussian algorithm (automatable)
 The golden ratio is a special relationship between two sizes, i.e. a number.If something is divided according to the golden ratio, it is called by most people aesthetically pleasing perceived.This number is \ (\ Phi = \ frac {1+ \ sqrt {5}} {2} \) or its reciprocal value \ (\ varphi = \ frac {\ sqrt {5} -1} {2} \).The number \ (\ Phi \) of the golden section is an irrational number.
 Longest side in a right triangle; is opposite to the right angle.

#### Indirect proportionality

Two sizes \ (x \) and \ (y \) are inversely proportional to each other, if a doubling, tripling, ... of the size \ (x \) results in halving, thirding, ... of the size \ (y \).

"The larger \ (x \), the smaller \ (y \)."

If there is indirect proportionality, then there is product equality:

\ (x \ cdot y = c \)

\ (c \) is the constant of proportionality and the following applies: \ (y = \ frac {c} {x} \).

A hyperbola results in the graphical representation of indirect proportionality.

 If you want to show that a statement \ (A \) results in a statement \ (B \), you can also go backwards this direct path.Because \ (A \ Rightarrow B \) is logically identical to \ (\ neg B \ Rightarrow \ neg A \).So one can alternatively show that from the non-occurrence of \ (B \) it follows that \ (A \) is not fulfilled.
 The irrational numbers are numbers that cannot be represented by fractions, that is, they are not contained in the rational numbers.Examples of irrational numbers are:The circle number \ (\ pi \)Euler's number \ (e \)The numbers of the golden ratio \ (\ Phi \) and \ (\ phi \)The square root of \ (2 \)
 Conic sections are curves that are obtained as a cross-section when you cut through a double cone with a plane.The conic sections include:PointcircleellipseparabolahyperboleConic sections describe the orbits of celestial bodies.
 A continued fraction is a term of the form\ (\ large b = a_0 + \ frac {1} {a_1 + \ frac {1} {a_2 + \ frac {1} {a_3 + \ ldots}}} \).With this formation law, a continued fraction can also be represented enumerating: \ (b = [a_0; a_1; a_2; a_3; \ ldots] \).If a continued fraction is infinitely long, then the number \ (b \) is not a rational number.
 For an angle \ (\ varphi \ ll 10 ° \) we have \ (\ tan \ varphi \ approx \ sin \ varphi \)
 multiplicative factor in front of a variableIn the term \ (3x ^ 2 \) 3 is the coefficient.
 Two geometric shapes are commensurable if they can be measured with the same measure - the unit.This means that the ratio of their sizes is rational.Commensurable surfaces are therefore in a relationship that is maintained a fraction of whole numbers expresses.
 "commutare" means "to exchange".Commutative law for addition:\ (a + b = b + a \)Commutative law for multiplication:\ (a \ cdot b = b \ cdot a \)Subtraction and division are non-commutative arithmetic operations.When exponentiating, only one commutative law applies to the exponents:\ (2 ^ 3 \ ne 3 ^ 2 \), but \ (\ left (a ^ 2 \ right) ^ 3 = a ^ 6 = \ left (a ^ 3 \ right) ^ 2 \).
 "Congruent"means"congruent".Preserve rotation, translation and mirroring Lengths and angles and thus form the Congruence maps. Two triangles are congruent to each other if one of the congruence theorems is fulfilled.
 Two triangles are congruent if one of the following conditions is true:SSS - The triangles coincide in all lengths of the sides.SWS - The triangles coincide in two side lengths and the intermediate angle.WSW - The triangles coincide at two angles and one side length.SsW - The triangles coincide in two side lengths and the angle on the smaller of the two sides.
 The cosine of an angle is a number that is assigned to that angle. This number is the aspect ratio of adjacent to hypotenuse in a right triangle.In the right-angled triangle \ (\ triangle ABC \) with a right angle at \ (C \), the cosine of \ (\ alpha \) corresponds to the ratio of adjacent \ (b \) to hypotenuse \ (c \). \ (\ cos \ alpha = \ frac {b} {c} \)The amount of the sine is a maximum of 1.
 The circle number \ (\ large \ pi \) is the constant of proportionality between the circumference and diameter of a circle.\ (U = \ pi \ cdot d \) or \ (\ pi = \ frac {U} {d} \)The following applies: \ (\ pi \ approx 3.141592653589793238462643383279502884197 \)

#### Laplace probability

A random experiment has finitely many possible outcomes that all equally likely then the following applies to the probability of event \ (A \): \ (P (A) = \ frac {\ # A} {\ # \ Omega} = \ frac {\ text {number of favorable results}} {\ text {Number of possible results}} \)

 Coefficient of the highest power of a variable in a termIn the term \ (3x ^ 5 + 4x ^ 3 + 2x ^ 2 + 5x-4 \) 3 is the leading coefficient.
 For quadratic equations of the form \ (ax ^ 2 + bx + c = 0 \) the solution formula applies\ (\ large x_ {1/2} = \ frac {-b \ pm \ sqrt {b ^ 2-4ac}} {2a} \)If one understands the term \ (ax ^ 2 + bx + c \) as a function term, this formula supplies the zeros of the parabola.The x-coordinate of the vertex of the parabola can also be read off: \ (x_S = \ frac {-b} {2a} \).
 Number of elements in a set \ (M \).Abbreviation: \ (\ # M \) or \ (| M | \).Example: \ (M = \ lbrace a, b, c, d \ rbrace \), then \ (\ # M = | M | = 4 \).
 A zero is the intersection of the graph of a function with the x-axis.Zeroing is determined by setting the function term \ (= 0 \).
 A parabola is a curve and belongs to the family of conic sections.The points of a parabola always have the same distance from a fixed point and a straight line. The graph of a quadratic function is a parabola.A parabola has a special point, the vertex.
 Two straight lines or lines are called parallel if they have a common perpendicular. A rectangle whose opposite sides are parallel is called a parallelogram. The diagonals in the parallelogram bisect each other. Parallelograms are point-symmetrical to the diagonal intersection.A parallelogram is a special case of a trapezoid.Area: \ (\ small A = g \ cdot h \)
 A power is just an abbreviation for a multiple multiplication by the same size.\ (\ underbrace {a \ cdot a \ cdot a \ cdots a} _ {\ text {n-times}} = a ^ n \)A power \ (b ^ n \) consists of one Base\ (b \) and the Exponents\ (n \).When calculating with powers, the power laws must be observed!
 The power laws are the basic calculation rules for powers:\ (a ^ m \ cdot a ^ n = a ^ {m + n} \)\ (a ^ m \ div a ^ n = \ frac {a ^ m} {a ^ n} = a ^ {m-n} \)\ (\ frac {1} {a ^ n} = a ^ {- n} \)\ (\ left (a ^ m \ right) ^ n = a ^ {m \ cdot n} \)Warning: these rules are only with multiplications or divisions applicable!
 Set \ (\ mathcal {P} (M) \) of all subsets of another set \ (M \).If the set \ (M \) \ (n \) has elements, then the power set has the power \ (2 ^ n \).
 Proportionality describes a simple connection between two quantities, which can be expressed by a "the more ..." sentence.One differentiatesdirect proportionality andindirect proportionality.In every form of proportionality there is an invariant quantity, the proportionality constant.
 actually "from a hundred"is an abbreviation for hundredths of a numberExample: \ (0.54 = 54 \% \).See also:percentagepercentage difference
 a fraction, the value of which is given as a percentage.At this break the Percentage\ (P \) by the so-called. Core value\ (G \) (reference point) divided.The following applies: \ (\ large p = \ frac {P} {G} = \ ldots \% \)
 denotes the specification of a difference one Percentage value\ (P \) to a Core value\(G\)in relation to the basic value.\ (\ large p = \ frac {P-G} {G} = \ frac {P} {G} -100 \% = \ frac {P} {G} -1 \)
 An image is called a point reflection at a center, ifthe center is the fixed point of the figure andeach point forms a segment with its image point, which is bisected by the center. To solve a quadratic equation, a creative zero is inserted to form a binomial formula.\ (\ begin {eqnarray} \ large ax ^ 2 + bx + c & = & a \ left (x ^ 2 + \ frac {b} {a} x \ underbrace {+ \ left (\ frac {b} {2a } \ right) ^ 2- \ left (\ frac {b} {2a} \ right) ^ 2} _ {= 0} \ right) + c \ & = & a \ left (x + \ frac {b} {2a } \ right) ^ 2- \ frac {b ^ 2} {4a} + ac \ end {eqnarray} \)The quadratic addition also serves to convert the function term of a quadratic function from the normal form to the vertex form.
 \ (\ large ax ^ 2 + bx + c = 0 \)is a quadratic equation. Such equations can be solved by completing the square or the solution formula.A quadratic equation can Have solution.
 A square root is a mathematical operation that gives, for a given area, the edge length of a square with this area.The square root is also known as the radix. The number under the root, as a radicand.Example: \ (\ sqrt {64} = 8 \), since \ (8 \ cdot 8 = 64 \). The square roots of numbers that are not squares of rational numbers are irrational numbers.
 The radical is the term that comes under a root.literally: that which is to be rootedExample:For \ (\ sqrt {a} \) \ (a \) is the radicand.Numerical example:For \ (\ sqrt {28} \) \ (28 \) is the radicand.
 A rational number is a number that can be represented as a fraction of an integer.Such a number can then be written either as a finite decimal fraction or as an infinite, periodic decimal fraction.The set of rational numbers is called \ (\ Large \ mathbb {Q} \).\ (\ mathbb {Q} = \ left \ lbrace \ frac {p} {q} | p \ in \ mathbb {Z} \ wedge q \ in \ mathbb {N} \ right \ rbrace \)The following applies: \ (\ mathbb {Q} \ supset \ mathbb {Z} \ supset \ mathbb {N} \).
 A diamond is a square with all sides of equal length. The diagonals are perpendicular to each other and form the axes of symmetry.A diamond is a special case of a parallelogram and a dragon square.
 Every real number can be written as an infinite decimal fraction.A real number is one element of the set of real numbersThe set of real numbers \ (\ Large \ mathbb {R} \) includes the rational and irrational numbers.The following applies: \ (\ mathbb {R} \ supset \ mathbb {Q} \ supset \ mathbb {Z} \ supset \ mathbb {N} \)The real numbers can be identified with a number line that has no gaps. In this case one speaks of a continuum.
 Technical term for a rotation of a geometric figure or a geometric body.