Term
| what the converse of Pythagorean Theorem says |
|
Definition
| If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. |
|
|
Term
| In a 30-60-90 triangle, (relative length of sides) |
|
Definition
the length of the hypotenuse is twice the length of the shorter leg and the
length of the longer leg is sqrt(3) times the shorter leg. |
|
|
Term
|
Definition
For any right triangle with legs a and b and hypotenuse c, the following relationship is
satisfied: a^2 + b^2 = c^2 |
|
|
Term
|
Definition
| the length of the hypotenuse is sqrt(2)times the shorter leg. |
|
|
Term
|
Definition
| one-half the product of its base and its height: A = bh/2 |
|
|
Term
| Side-Side-Side (SSS) postulate |
|
Definition
If three sides of one triangle are congruent to three sides of another triangle, then the
two triangles are congruent. |
|
|
Term
| Side-Angle-Side (SAS) postulate |
|
Definition
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the two triangles are congruent. |
|
|
Term
|
Definition
a triangle that includes only one right angle. the side opposite the right angle is the hypotenuse, and the
other two sides are the legs. |
|
|
Term
|
Definition
| If the hypotenuse and one leg of a right triangle are congruent to corresponding sides of another right triangle, then the triangles are congruent |
|
|
Term
| Two sides of a triangle are congruent if and only if |
|
Definition
| ... the angles opposite those sides are congruent |
|
|
Term
| An altitude of a triangle |
|
Definition
An altitude of a triangle is the segment from any vertex that is perpendicular to the
line containing the opposite side |
|
|
Term
| A triangle is called an acute, right, or obtuse triangle according to whether it includes |
|
Definition
| three acute angles, one right angle, or one obtuse angle, respectively |
|
|
Term
| Two lines that are not coplanar are called this |
|
Definition
|
|
Term
|
Definition
| a quadrilateral in which exactly one pair of sides is parallel |
|
|
Term
| A line segment with endpoints that lie on two nonconsecutive vertices of the polygon |
|
Definition
|
|
Term
| Two triangles are congruent if... |
|
Definition
| ... there is a correspondence between them such that every pair of corresponding sides is congruent and every pair of angles is congruent |
|
|
Term
| The Triangle Angle Sum Theorem |
|
Definition
| this states that the sum of the interior angles of a triangle is 180° |
|
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Term
|
Definition
| This means that 2 points are the same line |
|
|
Term
|
Definition
| a quadrilateral with exactly one pair of parallel sides |
|
|
Term
| The parallel postulate states: |
|
Definition
| Given a point not on a line, there is exactly one line parallel to the given line containing that point. |
|
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Term
|
Definition
| they are polygons with four sides |
|
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Term
| to show that two triangles are similar |
|
Definition
| One can use the AA similarity postulate, SAS similarity postulate, and SSS similarity postulate |
|
|
Term
| Two convex polygons are similar if |
|
Definition
there is a correspondence between their
vertices such that the corresponding angles are congruent and such that the ratios of the lengths of their corresponding sides are equal |
|
|
Term
| The perpendicular bisector of a line segment |
|
Definition
| the line perpendicular to the segment at its midpoint |
|
|
Term
| The area of a trapezoid is equal to... |
|
Definition
one-half the product of its height and the sum of the lengths of its
bases, b1 and b2: |
|
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Term
|
Definition
| a line segment with endpoints on the circle |
|
|
Term
If the measure of θ is 180°, then A and B are actually endpoints of a diameter and the two arcs are
semicircles |
|
Definition
| The measure of minor arc is the measure of its associated central angle, the measure of a semicircle is 180°, and the measure of a major arc is 360° minus the measure of the corresponding minor arc. |
|
|
Term
counting numbers or natural numbers
whole numbers
integers
Real numbers |
|
Definition
The ______________ are the numbers 1,2,3,4,....
The _____________ are the counting numbers plus zero.
The ________ are the whole numbers and all their opposites.
___________ consist of rational and irrational numbers |
|
|
Term
Rational numbers
Irrational numbers |
|
Definition
________________ include whole numbers, fractions, finite or repeating decimals, and percents.
_______________ are numbers that cannot be written as fractions since they are non-repeating and non-terminating decimals. |
|
|
Term
|
Definition
| these have the form "a + bi" |
|
|
Term
prime number
composite number |
|
Definition
A __________ is a number divisible only by 1 and itself.
A ________________ is a number with factors other than 1 and itself |
|
|
Term
|
Definition
| _______________ are the field of all rational and irrational numbers |
|
|
Term
|
Definition
| If B is a point of the arc AC, then m(arcAC) = m(arcAB) + m(arcBC). |
|
|
Term
|
Definition
| An angle is _________ in an arc if the sides of the angles contain the endpoints of the arc and if the vertex is a point on the circle touching the arc |
|
|
Term
|
Definition
| The measure of an _______________ is equal to half the measure of its intercepted arc |
|
|
Term
|
Definition
Two circles are _________ if they have congruent radii.
Two arcs are ____________ if they lie on the same circle (or on congruent circles) and have the same measure. |
|
|
Term
|
Definition
If two inscribed angles intercept the same arc or _________ arcs, then the angles are
_________.
Two arcs in the same circle or _________ circles are congruent if and only if their
corresponding chords are _________ |
|
|
Term
|
Definition
| A ______ of a circle is any line intersecting the circle at two points |
|
|
Term
|
Definition
| A _______ of a circle is any line that intersects the circle at exactly one point called the ________________ |
|
|
Term
|
Definition
| A ____ is tangent to a _______ if and only if the radius drawn to the point of tangency is perpendicular to the line |
|
|
Term
|
Definition
| The measure of an angle formed by two ______ that intersect in the interior of a circle is one half the sum of the ____ intercepted by the angle and by its opposite angle. |
|
|
Term
|
Definition
| The measure of an angle formed by an intersecting ______ and _______ is one half the difference of the intercepted arcs |
|
|
Term
|
Definition
| s = ar, where a is the arc measure and r is the radius of the circle |
|
|
Term
|
Definition
2(pi)r , where r is the radius of the circle
(convert from degrees to radians first) |
|
|
Term
| (x – h)^2 + (y – k)^2 = r^2 |
|
Definition
| The equation of a circle in the plane with center (h, k) and with radius r |
|
|
Term
| The standard form of the equation of a circle is... |
|
Definition
| x^2 + y^2 + Ax + By + C = 0 |
|
|
Term
|
Definition
| A _____ of a circle is a line segment with endpoints on the circle |
|
|
Term
|
Definition
| The __________ of a circle is length of a chord containing the center of the circle and is denoted by d. The ________ of a circle is twice its radius |
|
|
Term
|
Definition
| An ___ is any connected part of a circle |
|
|
Term
|
Definition
Any two distinct points A and B of the circle divide it into two arcs called the _____
arc AB and the _____ arc AXB |
|
|
Term
| (x - h)^2 + (y - k)^2 = r^2 |
|
Definition
| The equation of a circle in the plane with center (h, k) and with radius r |
|
|
Term
| Volume of a solid with trapezoidal bases and rectangular sides |
|
Definition
| (1/2)(b1 + b2)ah,where is the area of the trapezoid, and h is the height of the solid |
|
|
Term
| The area of a rectangle (and parallelogram) |
|
Definition
|
|
Term
|
Definition
|
|
Term
|
Definition
|
|
Term
|
Definition
|
|
Term
| The perimeter of a circle |
|
Definition
|
|
Term
|
Definition
|
|
Term
| Volume of a square pyramid |
|
Definition
| (1/3)s^2h, where s is the length of a side on the square base, s2 is the area of the square base, and h is the height of the pyramid |
|
|
Term
| Volume of a triangular pyramid |
|
Definition
| (1/3)(1/2ba)h, where (1/2ba) is the area of the triangular base, and h is the height of the pyramid |
|
|
Term
|
Definition
| (1/3)(pi^2)h, where pir^2 is the area of the circular base, and h is the height of the cone |
|
|
Term
| When a two-dimensional figure is changed in size by a factor of n |
|
Definition
| the area is changed by a factor of n^2 |
|
|
Term
|
Definition
lwh- for solids, this is measured in cubic units
• The volume of a non-pointed solid is found by multiplying the area of the base by the height of the solid.
• The volume of a pointed solid is found by multiplying the area of the base by the height of the solid and dividing by three
When the dimensions of a figure are increased by a factor of n, this is increased by a factor of n^3 |
|
|
Term
|
Definition
• For a solid, this is found by finding the sum of the areas of all of the surfaces of the solid.
• When the dimensions of a figure are increased by a factor of n, this is
increased by a factor of n^2 |
|
|
Term
|
Definition
a portion of a circle. The area of this can be found by setting up a proportion using the angle measure of the _______, the central angle of the circle, and the area of the entire circle.
The entire area of the circle would be 2(pi)r^2 |
|
|
Term
|
Definition
| This is a part of the circumference of a circle, so its length is proportion-ate in a way similar to the way a sector’s area is proportionate to its angle |
|
|
Term
| The volume of a non-pointed solid is found by... |
|
Definition
| ... multiplying the area of the base by the height of the solid |
|
|
Term
| The volume of a pointed solid is found by... |
|
Definition
| ... multiplying the area of the base by the height of the solid and dividing by 3 |
|
|
Term
| The surface area of a solid is found by... |
|
Definition
| ... finding the sum of the areas of all of the surfaces of the solid |
|
|
Term
| When the dimensions of a figure are increased by a factor of n, |
|
Definition
| ... any area is increased by a factor of n^2, and any volume is increased by a factor of n^3 |
|
|
Term
| How do you find the area of a sector of a circle? |
|
Definition
| proportions (ie- A/ (pi)r^2 = (central angle/ 360) |
|
|
Term
| How do you find arc lengths? |
|
Definition
| proportions (L/ 2(pi)r = #/ 360) |
|
|
Term
|
Definition
| Maps of states or drawings of buildings are drawn in this type |
|
|
Term
|
Definition
| This is a rectangular array of numbers (in rows and columns) |
|
|
Term
|
Definition
| the numbers in a matrix are referred to as the ________ of a matrix |
|
|
Term
| The dimensions of the matrix are said to be “m by n” |
|
Definition
which is written "m x n." An matrix has m rows and n columns. This notation
does not mean that you multiply m by n. |
|
|
Term
|
Definition
A matrix is sometimes called a row vector.
A matrix is sometimes called a column vector. |
|
|
Term
| adding & subtracting matices |
|
Definition
There are no surprises in addition and subtraction of matrices, however there is one
requirement. The matrices must have the same dimensions. Once you establish that the
matrices you are working with have the same dimensions, you can add or subtract each entry in
the first matrix with the corresponding entry in the second matrix. |
|
|
Term
|
Definition
| this changes the size of every entry by the same factor, called a scalar. A matrix of any dimensions may be multiplied by any scalar. |
|
|
Term
|
Definition
a rectangular array of #'s
the #'s in this are the ___________ of this |
|
|
Term
| rules for +'ing and -'ing matrices |
|
Definition
| no surprises but the dimensions must be equal |
|
|
Term
| 2 types of matrix multiplication |
|
Definition
scalar multiplication
multiplication of matrices |
|
|
Term
|
Definition
this type changes the size of every entry by the same factor in a matrix
(ex: r = 3) |
|
|
Term
| an (m x n) matrix can only be multiplied by an (n x p) matrix... |
|
Definition
| this means the # of columns of the 1st matrix must = the # of rows of the 2nd matrix |
|
|
Term
|
Definition
| if the 1st matrix has dimensions (m x n), and is multiplied by a 2nd matrix of dimensions (n x p), then the dimensions of the product matrix will be (m x p) |
|
|
Term
| how you multiply matrices |
|
Definition
if A = [a b] and B = [e f]
[c d] and [g h], then AB =
[a b] [e f] [ab + bg af + bh]
[c d] x [g h] = [ce + dg cf + dh] |
|
|
Term
| when multiplying, an identity matrix is... |
|
Definition
| ... a matrix whose entries are all 0's, except for the entries that lie on the main diagonal (top left to btm rgt). This for Multiplication is denoted as l |
|
|
Term
|
Definition
matrix algebra focuses on using the multiplicative inverse of a matrix. The inverse of A, denoted as A^(-1), is: AA^(-1) = l = A^(-1)A
To have an inverse, a matrix must be a square. Not all square matrices have an inverse |
|
|
Term
| inverse of a square matrix |
|
Definition
A^(-1) = 1/(ad - bc)[d -b]
[-c a] ad - bc =/ 0
(ad - bc) is called the determinant |
|
|
Term
|
Definition
a matrix is a rectangular array of #'s
Matrices can be +'d, -'d, and x'd
The associative property exists for addition and multiplication of matrices
The commutative property exists only in matrix addition
There are Identity Matrices in both addition and multiplication
Every matrix has an additive inverse |
|
|
Term
|
Definition
T^(-1) = [-2 3][a b] = [1 0]
[ 1 -2][c d] [0 1]
-2a + 3c = 1
-2b + 3d = 0
a - 2c = 0
b - 2d = 1,
a = -2, b= -3, c = -1, d = -2
--> T^(-1) = [-2 -3]
[-1 -2] |
|
|
Term
| row echelon form (of a matrix) |
|
Definition
* any form w/ all 0's at h btm o h matrix
* any row th has an entry othr tn 0 z h 1st non-zero entry
* any row th has 1 z h 1st non-zero entry has th entry frthr t h right tn h 1st non-0 entry o h row above |
|
|
Term
| The reduced row-echelon form, also known as the row reduced echelon form, of a matrix equation |
|
Definition
| requires back-substitution in order to develop the solution to the equation. |
|
|
Term
| The Gauss-Jordan elimination method |
|
Definition
while more complex and time-consuming
than reduced row-echelon method, this develops a form in which the solution is easy to identify and does not require back-substitution |
|
|
Term
| 2 ways to solve systems of linear equations |
|
Definition
reduced row-ecelon form
Gauss-Jordan elimination method |
|
|
Term
|
Definition
If a system of equations has no solution, it is called this
(ie: if the lines are parallel) |
|
|
Term
|
Definition
| If a system of equations has an infinite number of solutions, it is said to be this |
|
|
Term
|
Definition
|
|
Term
|
Definition
| this is a matrix that has either a single column or a single row. |
|
|
Term
Law of Cosines
says that for any triangle ABC) |
|
Definition
a^2 = b^2 + c^2 - 2bccosA
b^2 = a^2 + c^2 - 2accosB
c^2 = a^2 + b^2 - 2abcosC |
|
|
Term
the length of a vector,
u = [u1 u2 u3 ... uz], is denoted |u|... |
|
Definition
| |u| = sqrt( u1^2 + u2^2 + u3^2 + ... un^2) |
|
|
Term
| a Dot Product of 2 vectors, u = [u1 u2 u3 ... in] and v = [v1 v2 v3 ... vn]... |
|
Definition
| ... is denoted uv, and is defined uv = u1v1 + u2v2 + u3v3 + ... unvn |
|
|
Term
| (in other words) the dot product is.. |
|
Definition
the product of the lengths of the two vectors and the cosine of the angle between the vectors
cos? = (uv)/(|u||v|) |
|
|
Term
| a Dot Product of 2 vectors, u = [u1 u2 u3 ... in] and v = [v1 v2 v3 ... vn]... |
|
Definition
| ... is denoted uv, and is defined uv = u1v1 + u2v2 + u3v3 + ... unvn |
|
|
Term
|
Definition
[ i j k]
the determinant [u1 u2 u3]
[v1 v2 u3]
This is a vector. It is perpendicular to the plane on which the original two vectors lie. |
|
|
Term
| the length of the cross product u x v |
|
Definition
| the area of a parallelogram that has sides of u and v |
|
|
Term
the length of a cross product of
u x v x w |
|
Definition
| the volume of the parallelepiped created by u, v, and w |
|
|
Term
| the number of solutions of a system of linear equations |
|
Definition
this can be determined by graphing the system
this is also evident by the symbolic manipulations of the equations |
|
|
Term
| a three-variable linear system... |
|
Definition
... with 1 solution can be graphed as 3 planes meeting at exactly 1 point
... with no solution can be graphed as 3 planes that never coincide at the same time
... with an infinite # of solutions can be graphed as 2 or 3 planes that meet at a line |
|
|
Term
| a two-variable linear system |
|
Definition
| ... with one solution can be graphed as 2 lines that meet at exactly one point |
|
|
Term
| a two-variable linear system |
|
Definition
| with no solution can be graphed as parallel lines |
|
|
Term
| a two-variable linear system... |
|
Definition
| ... w/ an infinite # of solutions is graphed as 1 line b/c both equations describe the same line |
|
|
Term
|
Definition
| these can be used in a # of geometric applications |
|
|
Term
|
Definition
| between 2 vectors this produces a scalar |
|
|
Term
|
Definition
| between 2 vectors this produces a scalar |
|
|
Term
|
Definition
| between 2 vectors this produces a vector |
|
|
Term
|
Definition
| this is sometimes referred to as pre-calculus b/c many of its concepts serve as precursors to derivatives, integrals, and rates of change (Calculus) |
|
|
Term
| Trigonometric ratios (3 basic) |
|
Definition
|
|
Term
|
Definition
a circle with a radius of 1 (radius = 1)
legs = x (adjacent to angle @) and
y (opposite)
hypotenuse = 1 |
|
|
Term
|
Definition
1
@ = 0, coordinates (1, 0)
cos @ is x-coordinate (which = 1) |
|
|
Term
|
Definition
1
@ = 90⁰, coordinates (0, 1)
sin @ is y-coordinate (which = 1) |
|
|
Term
|
Definition
|
|
Term
|
Definition
@ = 270⁰, x- and y-coordinates (0, -1)
sin @ is y-coordinate |
|
|
Term
| Where is 405⁰ on a unit circle? |
|
Definition
405⁰ – 360⁰ = 45⁰
a unit circle is an easy way to show degrees and radians) |
|
|
Term
|
Definition
|
|
Term
|
Definition
|
|
Term
|
Definition
| cotangent, secant, cosecant |
|
|
Term
|
Definition
tan @ = sin @/ cos @
cot @ = cos @/ sin @ |
|
|
Term
|
Definition
cot @ = adjacent/ opposite
sec @ = hypotenuse/ adjacent
csc @ = hypotenuse/ opposite |
|
|
Term
| Pythagorean trigonometric identity |
|
Definition
* cos^2x + sin^2x = 1
* 1 + tan^2x = sec^2x
* cot^2x + 1= csc^2x |
|
|
Term
|
Definition
the height of each peak, or highest point, in a wave pattern measured from the middle of the wave. Equivalently, it is half the vertical height from lowest point, or trough to the peak.
A in the formulas f(t) = A sin (Bt + C) and f(t) = A cos (Bt+ C) |
|
|
Term
| range of a Cosine function |
|
Definition
|
|
Term
|
Definition
the number of wave patterns within a distance from 0 to 2(pi)
(the period and this are reciprocals)
In y = Asin(Bt + C), B is this |
|
|
Term
|
Definition
the horizontal distance on the x-axis between corresponding portions of the wave– from peak to peak, or trough to trough.
In other words, this shows the distance from where a wave pattern starts to where the wave pattern begins to repeat itself |
|
|
Term
|
Definition
a horizontal shift of a line on a graph (L or R)
(the amplitude, period, and frequency are unaffected)
- C (y= A sin (Bt + C)
B |
|
|
Term
|
Definition
a function that always holds true for
f(-x) = -f(x) |
|
|
Term
|
Definition
a function that always holds true for
f(-x) = f(x)
(this is symmetric around the y-axis) |
|
|
Term
|
Definition
|
|
Term
|
Definition
|
|
Term
| the tangent function is an odd function |
|
Definition
The tangent graph is not symmetrical over the y-axis, tan(-@) = -tan@
(A line that is tangent to a function with respect to the x-axis can also be referred to as the slope) |
|
|
Term
|
Definition
|
|
Term
| the graph of the secant function... |
|
Definition
like those of the tangent and cotangent functions, has undefined points at 90⁰ and -90⁰ (pi/2 & -pi/2)
symmetric about the y-axis |
|
|
Term
| the graph of the cosecant function... |
|
Definition
undefined at 0, pi, and -pi
not symmetric about the y-axis, so it is an odd function |
|
|
Term
| inverse trigonometric functions |
|
Definition
sin-1@, cos-1@, tan-1@, or
arcsin@, arccos@, arctan@
(NOT the reciprocals) |
|
|
Term
| the sine and cosine addition formulas |
|
Definition
sin (a + b) = sin(a)cos(b) + cos(a) + sin(b)
cos (a + b) = cos(a)cos(b) - sin(a) + sin(b)
sin (a - b) = sin(a)cos(b) - sin(b) + cos(a)
cos (a - b) = cos(a)cos(b) + sin(a) + sin(b) |
|
|
Term
| the addition formula for tangent |
|
Definition
tan(a + b) = (tan(a) + tan(b))/
(1 - tan(a)tan(b)) |
|
|
Term
| the subtraction formula for tangent |
|
Definition
tan(a - b) = (tan(a) - tan(b))/
(1 + tan(a)tan(b)) |
|
|
Term
|
Definition
(useful to find uncommon pts w/o a calculator)
sin(2a) = 2sin(a)cos(a)
cos(2a) = cos2a – sin2a – 2cos2a – 1= 1 – 2sin2a
tan(2a) = (2tan(a))/(1 – tan2a) |
|
|
Term
|
Definition
-sqrt(3)
tan(120⁰) = tan(2 x 60⁰)
tan(2a) = (2tan(a))/(1 – tan2a)
2 x sqrt(3)/(1 – (sqrt(3)^2)
2sqrt(3)/-2 |
|
|
Term
|
Definition
sin(x/2) = +/- sqrt((1 - cosx)/2)
cos(x/2) = +- sqrt((1 + cosx)/2)
tan(x/2) = +-sqrt((1 – cos x)/(1 + cosx)) |
|
|
Term
|
Definition
(sqrt(-(sqrt(3)-2)/2
= +/- sqrt((1 – cos(30⁰))/2)
30⁰ = sqrt(3)/2 →
sqrt(1 – (sqrt(3)/2)/2)
sqrt(-(sqrt(3)-2)/2 |
|
|
Term
|
Definition
| -1 <= (u * v)/ ||u||||v|| <= 1 |
|
|
Term
|
Definition
| coordinates in the form (x, y), such as (3, 3) |
|
|
Term
|
Definition
| coordinates in the form (r, @), where r is the distance from the origin to the point, & @ is the angle btwn the +'ve x-axis and the ray from the origin to the point |
|
|
Term
|
Definition
| equation for converting polar coordinates to rectangular coordinates |
|
|
Term
| polar coordinates (4, (pi)/3) into rectangular coordinates? |
|
Definition
(2, 2sqrt(3))
rcos@ = 4cos(pi/3) = 4 x ½ = 2
rsin@ = 4sin(pi/3) = 4 x sqrt(3)/2 =
2sqrt(3) |
|
|
Term
| (sqrt(x^2 + y^2), tan^-1(y/x)) = (r, @) |
|
Definition
| equation for converting regular coordinates into polar coordinates |
|
|
Term
changing rectangular coordinates
(2, 2) into polar coordinates |
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Definition
(sqrt(8), pi/4)
(sqrt(x^2 + y^2), tan^-1(y/x)) = (r, @)
sqrt(22 + 22) = sqrt(8)
tan^-1(y/x)) = tan^-1(1)) = pi/ 4 |
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Term
| Converting equation y = 3x + 2 into polar form? |
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Definition
r = 2/ (sin@ - 3cos@)
Since y = rsin@ & x = rcos@,
rsin@ = 3(rcos@) + 2
rsin@ = 3rcos@ + 2
rsin@ - 3rcos@ = 2
r(sin@ - 3cos@) = 2
r = 2/ (sin@ - 3cos@) |
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Term
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Definition
numbers which contain both a real part and an imaginary part. (ex- 3 + 7i)
(z is commonly used to denote these)
z = x + yi |
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Term
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Definition
| scalar multiples of i (sqrt(-1)) |
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Term
| equation for complex #'s in polar form |
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Definition
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Term
| z = 5 – 5i in polar form? |
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Definition
z = 5sqrt(2)(cos(7pi/4) + isin(7pi/4))
(z = 5 – 5i)
This point will be in the 4th quadrant of the complex plane, and since the x- and y-coordinates are both equidistant from the origin, the angle must be (7pi/4). We also know that r = sqrt(x^2 + y^2), so r = sqrt(50) = 5sqrt(2). So the polar form of the complex number is therefore z = (cos + i sin ). |
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Term
| What is y = (3i + 2)(6i – 2) in polar form? |
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Definition
r = (6i – 22)/ sin@ -->
y = -18 – 6i + 12i – 4 → y = 6i – 22
(x = rcos@ and y = rsin@)
rsin@ = 6i – 22 |
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Term
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Definition
if z = (rcos@) + (rsin@)i, then z^N = r^z(cos(n@) + isin(n@)
(useful when trying to find the powers of complex numbers and to simplify complex numbers.) |
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Term
| if z = (rcos@) + (rsin@)i, where r = 4, and @ = 60°, what is z^2? |
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Definition
8isqrt(3) - 8
By DeMoivre’s Theorem, we know that
z^2 = r^2(cos(2@) + isin(2@). And since r = 4, and @ = 60°, we can use substitution to get z^2 = 4^2(cos(2 60°) + isin(2 60°)= 16(cos(cos(2 60°) + isin(2 60°). Using the double angle formula cos(2@) = 1 - 2sin2@, we know that cos(2 60°) = 1 - 2sin2@60 = 1 – 2(sqrt(3)/2)2 = 1 – 2(3/4) = - ½
And using the double angle formula
sin(2@) = 2sin@cos@, we find that sin(2 60°) = 2sin(60°)cos(60°)=
2sqrt(3)/2 x ½ = sqrt(3)/2.
So using substitution, we get
16(-1/2 + i(sqrt(3)/2)=
8isqrt(3) – 8 |
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Term
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Definition
| the set of all possible outcomes |
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Term
| The formula for permutations |
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Definition
nPr= n!/(n-r)!
This formula describes the number of ways to arrange r elements out of n elements. |
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Term
| There are 36 people in the 6th grade class. How many different ways can the teacher line up 14 students? |
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Definition
36 x 35 x 34 ... 23
(36) P (14) = 36!/ (36 - 14) = 36!/22!=
36 x 35 ... 23 x 22!/ 22! = 36 x 35 ... 23 |
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Term
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Definition
! operation (ex- 3! = 3 x 2 x 1= 6)
n! = n * (n-1)(n-2)(n-3)...
(only works for nonnegative #'s)
note: 0! = 1 |
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Term
| Suppose that a license plate consists of two letters followed by four digits. The plates use only capital letters, and only the numbers 0 hrtough 9 (these letters and numbers can repeat; for example, a license plate might feature more than one 3 or more than one A). What is the total number of unique license plates possible? |
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Definition
6,760,000
__ __ __ __ __ __
L L # # # #
26 x 26 x 10 x 10 x 10 x 10 |
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Term
| fundamental counting principle |
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Definition
| the total number of possible outcomes following from a series of events is determined by multiplying all the ways that each individual event can occur |
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Term
| When choosing one card at random from a deck of cards, what is the probability of choosing a face card (i.e., a jack, queen, or king)? There are 52 cards in a deck and 12 face cards total |
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Definition
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Term
| f 2 coins are flipped simultaneously, what is the probability that both will land tails-up? |
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Definition
E = both coins land with their tail sides facing up. To use the formula, determine the ratio of favorable outcomes to total outcomes (H: heads,
T: tails)- 4 total outcomes: HH, HT, TH, TT. Clearly, only one of these four outcomes provides the favorable outcome of two tails (TT), and thus P(E) = ¼. |
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Term
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Definition
| total # of possible outcomes |
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Term
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Definition
| (# of favorable occurrences)/ (total # of outcomes) |
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Term
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Definition
the study of how likely something is
to happen; more specifically, probability helps us decide how likely it is that a certain outcome will follow from an event.
If probability is 0, it is impossible
If probability is 1, it is guaranteed |
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Term
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Definition
(used to count elements in a sample space in which the order does not matter
(n)C(r) = n!/ (n - r)r! |
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Term
Two restaurants in town offer vegetarian plates and a variety of vegetables from which to choose.
Restaurant A offers 6 different vegetables and customers can choose 2 per plate; restaurant B offers 5
vegetables and lets customers choose 3 per plate. Which statement below is correct?
A 5 more vegetarian plates are possible at restaurant B than are possible at restaurant A.
B 3 more vegetarian plates are possible at restaurant A than are possible at restaurant B.
C 3 more vegetarian plates are possible at restaurant B than are possible at restaurant A.
D 5 more vegetarian plates are possible at restaurant A than are possible at restaurant B. |
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Definition
D
Since the order doesn't matter, use the combination formula:
(6)C(2) = 6!/ 4!2!= 3 x 5 = 15
(5)C(2) = 5!/ 3!2! = 5 x 2 = 10 |
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Term
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Definition
| any point along a curve in which the concavity changes from down to up or up to down (In other words, the point x = c is a point of inflection if f''(x) < 0 when x < 0 and f''(x) > 0 if x > c) |
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Term
| Newton's method for approximating the roots of a function |
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Definition
| x(n + 1) = x(n) - (x(n))/(x'(n)) |
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