
Trigonometry Brain Summary
Malcolm
E. Hays 28 October 2002 
This is a list of all the thoughts located in the Trigonometry Brain. Each thought is followed by a statement indicating the content associated by that thought. Thoughts followed by (jump) are jump thoughts, which are located on the left hand side of the central thought in the Brain matrix. Jump thoughts often take the user to entirely different sections of the Brain or provide reference information about the central thought. Fundamentals of Trigonometry (jump) – The
basics needed in order to be properly prepared for Trigonometry Trigonometry (jump) – Return to the entry thought of the
Trigonometry Brain Algebra – A list of concepts covered in the Algebra
Brain necessary for understanding trigonometry Cartesian Plane – A brief review of the Cartesian
Plane, including the Distance and Midpoint Formulas Functions – A brief review of the idea of mapping
from one set of items to another Graphs of Functions – How to visually represent a
function Horizontal and Vertical Shifts – Moving a function around in the Cartesian
Plane Real Numbers –
Introduction to different number sets used throughout Trigonometry Absolute Value – Introduction to the absolute value
operator and what it really means Ordering Real
Numbers – How real numbers relate to each other
with respect to the origin Analytic Geometry  Studying
an equation from its graph and studying a graph from its equation Conic Sections  Classification
of conic sections by the discriminant of a quadratic function Polar Equations of Conics (jump) – Describing a conic section in polar
coordinates Circles  Definition of a
circle and its related equation Ellipses  Definition of
ellipse and its related equation Hyperbolas  Definition of hyperbola and its related
equation Parabolas  Definition of a
parabola and its related equation Translation of Conics  Moving the graph
of a conic section around in the plane Circle Translation  How to move a circle around in the plane Ellipse Translation  How to move an ellipse around in the plane Hyperbola Translation  How to move a
hyperbola around in the plane Parabola Translation  How to move a
parabola around in the plane Lines – Introduction to the concept of a line and
its slope Plane Curves and Parametric Equations – Using a third variable to describe the
motion of an object through time Polar Coordinates – Introduction to an alternate method of
expressing functions in terms of radii and angles Graphing Polar
Equations – Demonstration on how to graph an equation
in polar coordinates Polar
Equations of Conics – Describing a conic section in polar
coordinates Conic
Sections (jump) – Classification
of conic sections by the discriminant of a quadratic function Rotation of Axes – Transforming a quadratic equation by
rotating the coordinate axes to eliminate the xyterm Analytic Trigonometry  Seconddegree
polynomial functions Inverse Trig Functions  Definition
of the inverse of a trigonometric function along with its graph Composition of
Trig Functions – Composing one trigonometric function with
another MultipleAngle Formulas – How to evaluate trigonometric functions
of nθ PowerReducing Formulas – Transforming trigonometric functions
raised to a power greater than 1 into first power expressions HalfAngle Formulas – Evaluating trigonometric functions of θ / 2 Sum and Difference Formulas – Evaluating trigonometric functions of θ ± φ ProducttoSum / SumtoProduct
Formulas – Transforming a sum of the trig functions of
two different angles into a product and vice versa Trigonometric Equations – Solving equations involving trigonometric
functions using primarily algebraic techniques Trigonometric Identities – Demonstration of how to relate
trigonometric functions to each other Six Basic Trig Functions (jump) – Definition of the sine, cosine, tangent,
cotangent, secant, and cosecant functions Verifying Trig
Identities – Examples of how to determine if a
trigonometric identity is true or not
Applications of Trigonometry – The many
uses of trigonometry, such as the laws of sine and cosine Law of Cosines  Solving
an oblique triangle given three sides (SSS) or two sides and their included
angle (SAS) Herons's Area Formula – Using the law of cosines to find the area
of an oblique triangle Law of Sines – Solving an oblique triangle given two
angles and any side (AAS or ASA) or two sides and an angle opposite one of the
sides (SSA) Area of Oblique
Triangle – Using
the law of sines to find the area of an oblique triangles Law of Tangents – An
obscure law derived from the law of sines and some trigonometric identities Vectors – A
number that has both a magnitude and
a direction Component Form – One of the several ways in which we
represent vectors, handy for many applications Vector
Operations – Scalar multiplication and vector addition Cross
Product – A special form of vector multiplication
that yields a third vector perpendicular to the first two Direction
Angles – The angle in which the vector u
points relative to the positive xaxis Dot
Product – A special form of vector multiplication
that yields a directionless real number as a result Angle
Between Two Vectors – Given two vectors, find the angle between
them Finding
Vector Components – Projecting a vector onto a set of
coordinate axes to find the component of that vector Work – A
primary application of projecting vectors to find the amount of work done on
an object based on the forces acting on that object Unit
Vectors – A handy form of notation that indicates the
components of a vector relative to a set of coordinate axes Complex Numbers – Numbers that contain both a real part and
an imaginary part Complex Plane (jump) – Using a set of coordinate axes to
represent a complex number Trigonometric Form – Writing a complex number in terms of the
trigonometric functions DeMoivre's Theorem – Raising a complex number in trig form to a
power greater than 1 Multiplication and
Division – Multiplying and dividing complex numbers in
trigonometric form Roots of Complex
Numbers – A formula to find the nth roots of a complex number Exponential and Logarithmic Functions  Basic
concepts of functions based on the number e Exponential Functions  Basic definition
of the exponential function Properties of Exponents (jump) – The
various rules governing the behavior of exponents Exponential and Logarithmic Equations – How to use logarithms to solve mathematical
equations Logarithmic/Exponential Models  Applications of
logarithmic/exponential functions Exponential Growth and Decay  Very common application of logs Gaussian  Model which produces the bellshaped
curve used in statistical analysis Logarithmic  Used in a wide variety of
applications including earthquakes, sound, and time of death Logistics Growth  Model used to accurately
represent population growth in an environment Logarithmic Functions – The inverse of an exponential function Properties of Logarithms (jump)  Various rules
governing behavior of logarithmic functions Exponential / Logarithmic Equations – How to use logarithms to solve
mathematical equations Logarithmic/Exponential Models  Applications of
logarithmic/exponential functions Exponential Growth and Decay  Very common application of logs Gaussian  Model which produces the
bellshaped curve used in statistical analysis Logarithmic  Used in a wide variety of
applications including earthquakes, sound, and time of death Logistics Growth  Model used to accurately
represent population growth in an environment Natural Base e  Definition of
logarithm of base e Trigonometric Functions – The fundamental basis of trigonometry Angles – Definition of an angle in terms of
radians and degrees, and conversion from one to the other Reference Angles – The acute angle formed by the terminal
side of an angle and the nearest xaxis Six Basic Trig Functions – Definition
of the sine, cosine, tangent, cotangent, secant, and cosecant functions Trigonometric Identities (jump) – Demonstration of how to relate
trigonometric functions to each other Unit Circle (jump) – A useful
device for evaluating trigonometric functions of certain angles Applications – Several applications of the six basic
trigonometric functions Bearings – A
means of finding one's way in the world Harmonic Motion – Anything that vibrates or oscillates
falls into this category Right Triangle Trigonometry – Finding the missing components of a right
triangle given at least two other components Graphs of
Trig Functions – The visual
representations of the six basic trigonometric functions Cotangent Curves – Graphing the cotangent function Amplitude
and Period (jump) – The "height" and
"frequency" of a given trigonometric curve Phase Shift (jump) – Moving a trigonometric function from its
standard position along the xaxis Damped Trigonometric Graphs – Introducing additional terms in a
trigonometric equation affects the curve Reciprocal Curves – Graphs of the secant and cosecant
functions Sine and Cosine Curves – Graphs of the sine and cosine functions Amplitude
and Period – The "height"
and "frequency" of a given trigonometric curve Cotangent Curves
(jump) – Graphing
the cotangent function Tangent Curves (jump) – Graphing the tangent function Phase
Shift – Moving a trigonometric
function from its standard position along the xaxis Cotangent Curves
(jump) – Graphing
the cotangent function Tangent Curves (jump) – Graphing the tangent function Tangent Curves – Graphing the tangent function Amplitude
and Period (jump) – The "height" and "frequency"
of a given trigonometric curve Phase Shift (jump) – Moving a trigonometric function from its
standard position along the xaxis 
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