Domain: describes all the values of that ‘x’ is able to take. Range: describes all the values that ‘y’ is able to take. Period: of a graph describes the part of a graph, which is periodically repeated. It uses the horizontal scale. Amplitude: of a ‘wave’ graph is the height of the wave fro the horizontal.
Relations Between the Trig Ratios sin θ = y tan θ = y/x Pythagorean Identities 1. tan θ = sin θ /cos θ 2. cot θ = cos θ /sin θ 3. cos² θ + sin² θ = 1 4. 1 + tan² θ = sec² θ 5. cot² θ + 1 = cosec² θ cos θ = x cot θ = x/y
Mathematics Summary Sheets – Trigonometry
2
Trig Equations We need: • • Domain: sinx cosx all real x all real x
Range: –1 ≤ sinx ≤ 1 1 ≤ cosx ≤ 1 ∞ < tanx < ∞
• • • •
tanθ exists except for θ = ± 90°, ± 270°, ± 450° … ASTC results Special triangles (exact ratios) Results for 0°, 90°, 180°, 270°, 360°. B c A a C
Area of a triangle
A= ½absinC
Sine Rule a b c ______ = ______ = ______ sinA sinB sinC
...Teaching trigonometry using Empirical Modelling
0303417
Abstract
The trigonometric functions sin(x), cos(x) and tan(x) are relationships that exist between the angles
and length of sides in a rightangled triangle. In Empirical Modelling terms, the angles in a triangle
and the length of the sides are observables, and the functions that connect them are the definitions.
These welldefined geometric relationships can be useful when teaching GCSElevel students about
the functions, as they provide a way to visualise what can be thought of as fairly abstract functions.
This paper looks at how different learning styles apply to Empirical Modelling, and presents a practical example of their use in a model to teach trigonometry.
1 Introduction
The trigonometric functions sin(x), cos(x) and tan(x)
are relationships that exist between the angles and
length of sides in a rightangled triangle. In Empirical Modelling terms, the angles in a triangle and the
length of the sides are observables, and the functions
that connect them are the definitions. These welldefined geometric relationships can be useful when
teaching GCSElevel students about the functions,
as they provide a way to visualise what can be
thought of as fairly abstract functions. Rather than
teaching students by showing them diagrams in an
instructive way (already a good way of doing it), a
constructive approach may allow students to gain a
better understanding...
...Right Triangle TrigonometryTrigonometry is a branch of mathematics involving the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Also the ability to use and manipulate trigonometric functions is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Rightangled Triangles In a rightangled triangle the three sides are given special names. The side opposite the right angle is called the hypotenuse (h) – this is always the longest side of the triangle. The other two sides are named in relation to another known angle (or an unknown angle under consideration).
If this angle is known or under consideration
h
θ
this side is called the opposite side because it is opposite the angle
This side is called the adjacent side because it is adjacent to or near the angle Trigonometric Ratios In a rightangled triangle the following ratios are defined sin θ = opposite side length o = hypotenuse length h cosineθ = adjacent side length a = hypotenuse length h
tangentθ =
opposite side length o = adjacent side length a
where θ is the angle as shown
These ratios are abbreviated to sinθ, cosθ, and tanθ respectively. A useful memory aid is Soh Cah Toa pronounced ‘socartowa’
Page 1 of 5
Unknown sides and angles in right angled triangles can be found using these ratios. Examples Find the value of the indicated unknown (side...
...Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.
Contents
f one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined,...
...Trigonometry (from Greek trigōnon "triangle" + metron"measure"[1]) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.


\History
Main article: History of trigonometry
The first trigonometric tablewas apparently compiled byHipparchus, who is now consequently known as "the father of trigonometry."[3]
Sumerian astronomers introduced angle measure, using a division of...
...Trigonometry
 Introduction to trigonometryAs you see, the word itself refers to three angles  a reference to triangles. Trigonometry is primarily a branch of mathematics that deals with triangles, mostly right triangles. In particular the ratios and relationships between the triangle's sides and angles. It has two main ways of being used: 1. In geometryIn its geometry application, it is mainly used to solve triangles, usually right triangles. That is, given some angles and side lengths, we can find some or all the others. For example, in the figure below, knowing the height of the tree and the angle made when we look up at its top, we can calculate how far away it is (CB). (Using our full toolbox, we can actually calculate all three sides and all three angles of the right triangle ABC). 2. AnalyticallyIn a more advanced use, the trigonometric ratios such as as Sine and Tangent, are used as functions in equations and are manipulated using algebra. In this way, it has many engineering applications such as electronic circuits and mechanical engineering. In this analytical application, it deals with angles drawn on a coordinate plane, and can be used to analyze things like motion and waves. Chapter1Angles in the Quadrants( Some basic Concepts)In trigonometry, an angle is drawn in what is called the "standard position". The vertex of the angle is on the origin, and one side of the angle is fixed and drawn along the positive...
...ENGTRIG: LECTURE # 4.2 Spherical Trigonometry
Spherical Trigonometry
Engr. Christian Pangilinan
Areas of a Spherical Triangle
A=
π R2 E
180o E R
E = A + B + C − 180o
Where:
spherical excess radius of the sphere
Spherical Triangles Part of the surface of the sphere bounded by three arcs of three great circles Right Spherical Triangle – a spherical triangle containing at least one right angle
If the sides are known instead of the angles, then L’Huiller’s Formula can be used to solve for the spherical excess
1 s s − a s − b s−c tan E = tan tan tan tan 2 2 2 2 2 a+b+c Where: s= 2
Solutions to Right Spherical Triangles (C = 90o)
Sides a, b, c are based on the corresponding arc lengths S = rθ that is based on its corresponding interior angles S a = rθ BOC ; Sb = rθ AOC ; Sc = rθ AOB and that a + b + c < 360o Angles A, B, C of a spherical triangle are measured by the corresponding dihedral angles of the trihedral angle A : B − OA − C ; B : A − OB − C and
C : A − OC − B
Napier’s Rules: 1. The sine of any middle part is equal to the product of the tangents of the adjacent parts 2. The sine of any middle part is equal to the product of the opposite parts *“co” indicates complement
s i n ( middle ) = product of t a n ( adjacent ) s i n ( middle ) = product of c o s ( opposite )

Angles are measured between the tangents at the point (A) to the arcs of the other points...
...Physics
Glossary
Electric circuit  one simple, complete conducting circuit pathway
Electronic gadget  a machine that consists of multiple circuits and transducers
Transducer – device that converts energy from one form to another
Input transducer – converts other forms of energy (sound, light, heat) into electrical energy, e.g. microphone
Output transducer – converts electrical energy into other forms of energy, e.g. speakers
Processor Component – found in electronic gadgets, receives signals from input transducer and responds by operating the output transducer
Transistor – device found in processors that can amplify electrical signals or act as a switch
Diodes – also found in processors, allows the electric current to flow in only one direction
Semiconductor – a material whose ability to conduct electricity is somewhere between a conductor and insulator
LED – a light emitting diode, which is a transistor which transforms electric current into thermal and light energy. Found in traffic lights, car lamps and indicator lights, and are sturdy, operate for a longer time and use less energy than standard lights.
Integrated circuit – a usually complex circuit with many components, and is packaged into a small unit called a chip
Magnetic field – is created when an electric charge moves; it is a region where a permanent experiences a magnetic force of attraction or repulsion
Electromagnet – is created when a coil of wire is wrapped around a soft piece of iron when...
...A Complicated Kindness SummarySheet
Core Mennonite Beliefs
* First and foremost: Christians
* Specific Mennonite philosophies
* Non violence
* No circumstances
* Mennonites are exempt from going to war
* Rejection of the world
* Should focus on heaven
* Things in real life are distractions
* Community
* Strong emphasis on doing charity and helping each other
* Putting faith into action
* Actions reflect innerself
Religious fundamentalism and the comingofage novel
* Coming of age
* The achievement of maturity
* Transition from childhood to adulthood
* Fundamentalist religions have strict code for adult behavior
* Comingofage novels set in fundamentalist communities often feature a protagonist who rebels against religious authority
* Comingofage novels set in fundamentalist communities tend to end either with:
* The protagonist coming to a more mature understanding of his/her religion OR
* The protagonist rejecting his/her religion
Significance of the songs
* Rejection of Mennonite values
* Yet some songs have Christian themes
* Embracing of secular world
* Connection to other characters
* Tash
* Trudie
* Ray
* Travis
* Other young people
Plot
* It does not exist
* Wanders around, frequent flashbacks
* Lack of...