Double compound angle formulae
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This document provides rules and formulae for double angles in trigonometry. It contains: 1) Formulae for sin(A+B), cos(A+B), sin2A, and cos2A in terms of sinA, cosA, sinB, and cosB. 2) Examples of using the double angle formulae to find sin2A and cos2A when given sinA and cosA. 3) Alternative versions of the cos2A formula as cos2A = 1 - 2sin2A or cos2A = 2cos2A - 1, which are useful for trigonometric equations.
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SQL Assessment Command Statements
The document provides examples and explanations of common SQL commands used to query, manipulate, and manage data in a relational database. Key points include:
- The SELECT statement is used to query data from database tables. The WHERE clause filters results.
- INSERT, UPDATE and DELETE statements are used to add, modify and remove rows in database tables.
- CREATE, ALTER, DROP statements manage the database schema by creating, modifying and deleting tables, columns, keys and constraints.
- PRIMARY KEY and FOREIGN KEY constraints enforce data integrity and relationships between tables.
The Rise and Fall of the Roman Empire
The document provides an overview of the rise and fall of the Roman Empire from 509 BCE to 476 CE. It discusses the transition from Roman Republic to Roman Empire under Augustus, key emperors like Tiberius, Caligula, Nero, Vespasian, and Hadrian, as well as the division of the empire into eastern and western halves under Diocletian. It also examines factors that contributed to the decline of the western empire, including invasions by barbarian tribes and the removal of the capital to Constantinople, while the eastern empire continued on as the Byzantine Empire until 1453.
National 5 Graphic Communication
The document discusses color theory and desktop publishing principles. It describes primary, secondary and tertiary colors. It also discusses tints, shades, warm and cold colors. It outlines design principles like alignment, bleed, captions and drop caps. It provides information on fonts, headers, footers, guides and page layout features.
Vector journeys!
The document discusses vectors and their components. It provides examples of finding the components of vectors using other known vectors, such as finding the components of vector AC given the components of vectors AB and BC. It also discusses using the triangle rule to find unknown vectors. Diagrams are included to illustrate different vector problems and solutions.
Vector multiplication dot product
The document discusses vector multiplication using the dot product. It provides two formulas for calculating the dot product:
1) a.b = |a| |b| cosθ, where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and cosθ gives the effectiveness of their interaction. The dot product is largest when the vectors are parallel (θ = 0).
2) For vectors in 3D space, a.b = x1x2 + y1y2 + z1z2, where the x, y, z components of each vector are multiplied and summed.
It provides examples of calculating the dot product using both formulas in different vector
Dot product calc angle to finish!
This document discusses using the dot product to calculate the angle between two vectors. It provides the formula for calculating the dot product and shows examples of using the formula to find the angle between vectors. It also discusses how if two vectors are perpendicular, their dot product will equal 0.
Unit vectors 14
This document introduces unit vectors and how they can be used to write the components of a vector. It defines a unit vector as having a magnitude of 1 and lists the standard unit vectors i, j, k. It then shows how any vector can be written as the sum of its components in the direction of each unit vector, using examples like 3i + 4j + 7k. Finally, it provides additional examples of writing vectors in component form using unit vectors and calculating the sum of two vectors.
Vector bits and pieces
This document provides information about vectors, including:
- The relationship between the magnitude of a vector and the cosine of the angle between two vectors in determining their dot product.
- Unit vectors have a magnitude of 1 and are perpendicular to each other.
- Examples are given of calculating the dot product of vectors in different geometric configurations by decomposing into the dot products of the vectors with the unit vectors.
- Key steps are to start from the same point when calculating dot products of vectors in a figure.
Vectors intro
Vectors can represent journeys with distance and direction. They have components that work like coordinates and magnitude that is calculated using Pythagoras. Vectors can be added by placing them nose to tail and adding corresponding components, subtracted by adding the negative vector, and multiplied by a scalar by multiplying each component.
Ratios
The document discusses applying ratios to vectors. It provides examples of using ratios to find unknown points along vectors. The key steps are: 1) Form an equation using the position vector rule relating the unknown point P to the known endpoints A and B; 2) Make P the subject of the equation; 3) Solve the equation to find the coordinates of P. Examples show setting up equations for points dividing vectors in given ratios and solving to find the coordinates of the unknown points.
Parallel + collinear vectors
Vectors AB and CD are parallel if their direction vectors are proportional.
Vectors A, B, and C are collinear if their direction vectors are proportional and they share a common point.
Three examples are given to demonstrate proving vectors are parallel or collinear by showing their direction vectors are proportional.
Position and 3 d vectors amended
This document discusses position vectors in 2D and 3D coordinate systems.
- A position vector starts at the origin and represents the displacement from the origin to a point P defined by its coordinates.
- The position vector rule states that the position vector from point P to point R is equal to the vector from R's position vector subtracted by P's position vector.
- Position vectors and their properties like magnitude, addition, subtraction, and scalar multiplication follow the same rules in 3D as in 2D. Several examples are provided to demonstrate calculating the position vector and magnitude between two points in 3D space.
Solving trig equations higher
The document provides information on solving more difficult trigonometric equations. It discusses finding the acute angle and related angles, using trig function properties to isolate the angle, and considering additional solutions that occur every period of the trig function. Example problems are worked through step-by-step to demonstrate these concepts.
Solving trig equations + double angle formulae
1) The document discusses solving trigonometric equations using compound angle formulas and double angle formulas.
2) It provides examples of using these formulas to solve equations such as sin2x + cosx = 0, 2sinxcosx + cosx = 0, and cos2x - sinx = 0.
3) The key steps are to use a double angle formula to get one side equal to zero, factorize the equation, and then solve the resulting trigonometric mini equations.
Solving exponential equations
This document discusses solving exponential equations by taking the log of both sides. It shows how to solve equations with unknown powers and equations containing e as the base. The key steps are:
1) Isolate the term with the unknown power
2) Take the log of both sides
3) Use properties of logs to isolate the unknown variable
4) Calculate the value of the variable
Examples are provided to illustrate solving equations with unknown powers, equations containing e, and applying the method to context problems. The overall approach is to take logs to transform exponential equations into linear equations that can be easily solved.
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Double compound angle formulae
- 1. Block 2 Double Compound Angle Formulae
- 2. What is to be learned? • Rules for double angles!!!!!
- 3. Sin(A + B) = Sin(A + A) = sin2A = SinA CosBSinA CosB + CosA SinB+ CosA SinB SinA CosASinA CosA + CosA SinA+ CosA SinA 2SinA CosA2SinA CosA
- 4. Cos(A + B) = Cos(A + A) = Cos2A = CosA CosBCosA CosB –– SinA SinBSinA SinB CosA CosACosA CosA –– SinA SinASinA SinA CosACosA –– SinASinA( )2 ( )22 2
- 5. Double Angle Formulae If we have two equal angles sin2A = Cos2A = 2SinA CosA2SinA CosA CosACosA( )2 ( )2 - SinA- SinA Cos ACos A –– Sin ASin A2 2 written as on formula sheet
- 6. cos A = 2 /√13 Find sin 2A and Cos 2A Sin2A = 2sinAcosA = 2 X Cos2A = Cos2 A – Sin2 A = = = A2 √133 sinA = 3 /√13 3 /√13 X 2 /√13 = 12 /13 (2 /√13)2 – (3 /√13)2 4 /13 – 9 /13 -5 /13
- 7. 1 – Sin2 A 1 – Sin2 A – Sin2 A CosCos22 AA –– SinSin22 AACos2A = CosCos22 A =A = Cos2A = = 1 – 2Sin2 A 1 – Cos2 ASinSin22 A =A = Cos2A = CosCos22 AA –– (1 – Cos2 A) = 2Cos= 2Cos22 A – 1A – 1
- 8. Cos2A = 1 – 2Sin2 A = 2Cos= 2Cos22 A – 1A – 1 Cos 2A Other versions and Cos2A handy for trig equations also on formula sheet