L33 Clarifying Examples and Activities2 - mi-math
... when constructing proofs. However, there are times when a direct reasoning path is hard to see, or when using indirect reasoning is more elegant. Students should see several proofs using indirect methods (proof by contradiction) over their high school mathematics experience. They should also be able ...
... when constructing proofs. However, there are times when a direct reasoning path is hard to see, or when using indirect reasoning is more elegant. Students should see several proofs using indirect methods (proof by contradiction) over their high school mathematics experience. They should also be able ...
Chapter 4 Euclidean Geometry
... there exists in the plane of ` and P one and only one line m that passes through P and is parallel to `. Theorem 4.4 If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. Proof: Assume to the contrary, then use the parallel postulate to draw a con ...
... there exists in the plane of ` and P one and only one line m that passes through P and is parallel to `. Theorem 4.4 If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. Proof: Assume to the contrary, then use the parallel postulate to draw a con ...
unit 6: right triangles and trigonometry unit 6: right triangles and
... SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles SRT.C.8: Use trigonometri ...
... SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles SRT.C.8: Use trigonometri ...
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.