Chapter 3 – Parallel and Perpendicular Lines
... Ex 3: The population of a small town was 32,150 in 1990. A census in 2002 showed that the population was 22,550. Find the average rate of change in population per year from 1990 to 2002. Assuming this rate remains constant, what should the population be in 2003? ...
... Ex 3: The population of a small town was 32,150 in 1990. A census in 2002 showed that the population was 22,550. Find the average rate of change in population per year from 1990 to 2002. Assuming this rate remains constant, what should the population be in 2003? ...
Assignments Intro Proofs
... READ: Remember from the last assignment: Numbers can always be added and subtracted. It makes no sense to add or subtract people. Line segments can sometimes be added or subtracted (if you don’t remember when, review the note after homework IP – 1 #5). Angles are like segments. They can sometimes b ...
... READ: Remember from the last assignment: Numbers can always be added and subtracted. It makes no sense to add or subtract people. Line segments can sometimes be added or subtracted (if you don’t remember when, review the note after homework IP – 1 #5). Angles are like segments. They can sometimes b ...
Trouncing Trig Technicalities
... As you can see, the first three letters in the full name make up the abbreviations, except for cosecant’s. ...
... As you can see, the first three letters in the full name make up the abbreviations, except for cosecant’s. ...
Similar Polygons
... Similar Polygons Two polygons with the same shape are called similar polygons. The symbol for “is similar to” is ∼. (Example: ABC EFG) (Notice that is a portion of the “is congruent to” symbol, ≅.) When two polygons are similar, these two facts both must be true: – Corresponding angles are eq ...
... Similar Polygons Two polygons with the same shape are called similar polygons. The symbol for “is similar to” is ∼. (Example: ABC EFG) (Notice that is a portion of the “is congruent to” symbol, ≅.) When two polygons are similar, these two facts both must be true: – Corresponding angles are eq ...
Document
... For Exercises 13–16, determine whether the parallelogram is a rhombus, a rectangle, or a square. Give the most precise description in each case. 13. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and ...
... For Exercises 13–16, determine whether the parallelogram is a rhombus, a rectangle, or a square. Give the most precise description in each case. 13. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and ...
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.