topic 11 triangles and polygons
... The figure has to be bounded by line segments (but the segments do not all have to be congruent), and the line segments can only intersect at their endpoints. The figure must be closed. (There has to be an interior and an exterior? How do we know which is the interior?) Only two segments can i ...
... The figure has to be bounded by line segments (but the segments do not all have to be congruent), and the line segments can only intersect at their endpoints. The figure must be closed. (There has to be an interior and an exterior? How do we know which is the interior?) Only two segments can i ...
Quadrilaterals
... scientific methodology. However, his teachings emphasized philosophical speculation over practical applications of science. It is therefore fitting that a legend tells of his falling into a well while pondering the heavens. A passing peasant is said to have fished him out with the comment "here is a ...
... scientific methodology. However, his teachings emphasized philosophical speculation over practical applications of science. It is therefore fitting that a legend tells of his falling into a well while pondering the heavens. A passing peasant is said to have fished him out with the comment "here is a ...
Trigonometric ratios of an angle of any size
... Knowledge of the trigonometrical ratios sine, cosine and tangent, is vital in very many fields of engineering, mathematics and physics. This unit explains how the sine, cosine and tangent of an arbitrarily sized angle can be found. In order to master the techniques explained here it is vital that you ...
... Knowledge of the trigonometrical ratios sine, cosine and tangent, is vital in very many fields of engineering, mathematics and physics. This unit explains how the sine, cosine and tangent of an arbitrarily sized angle can be found. In order to master the techniques explained here it is vital that you ...
Lesson 4.4 Powerpoint - peacock
... Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = x/1 and sin θ = y/1. So what if the radius (hypotenuse) is not 1? ...
... Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = x/1 and sin θ = y/1. So what if the radius (hypotenuse) is not 1? ...
topic 9 geometry proofs
... When you solve an equation in algebra, you are doing a step-by-step process to convince someone that an answer is valid. That is the process of writing a proof. So we will begin by looking at the steps in the solution of an equation, adding the algebra reasons for each step. These reasons will come ...
... When you solve an equation in algebra, you are doing a step-by-step process to convince someone that an answer is valid. That is the process of writing a proof. So we will begin by looking at the steps in the solution of an equation, adding the algebra reasons for each step. These reasons will come ...
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.