m3hsoln2.tex M3H SOLUTIONS 2. 3.2.2017 Q1 (Angle at centre
... ∠OCA = θ, ∠OBA = φ. So AB subtends ∠ACB = θ + φ at the circumference. In ∆AOC, ∠AOC = π − 2θ (angle sum is π), and similarly ∠BOC = π − 2φ. The three angles are O sum to 2π; the two just mentioned sum to 2π − 2θ − 2φ. So ∠AOC = 2(θ + φ) = 2.∠ACB. // Note that if the chord goes through the centre, th ...
... ∠OCA = θ, ∠OBA = φ. So AB subtends ∠ACB = θ + φ at the circumference. In ∆AOC, ∠AOC = π − 2θ (angle sum is π), and similarly ∠BOC = π − 2φ. The three angles are O sum to 2π; the two just mentioned sum to 2π − 2θ − 2φ. So ∠AOC = 2(θ + φ) = 2.∠ACB. // Note that if the chord goes through the centre, th ...
CHAPTER 2: MATH NOTES Angle Relationships Naming Parts of
... sides of the right triangle (the sides that meet at the right angle) are called the legs of the triangle and the longest side (the side opposite the right angle) is called the hypotenuse of the triangle. ...
... sides of the right triangle (the sides that meet at the right angle) are called the legs of the triangle and the longest side (the side opposite the right angle) is called the hypotenuse of the triangle. ...
Grade 8 Pre-Algebra Curriculum
... In Grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles to find the measures of unknown angles. This unit extends that knowledge to angle relationships that are formed when two parallel lines are cut by a transversal and why the exterior angles of a triangle ...
... In Grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles to find the measures of unknown angles. This unit extends that knowledge to angle relationships that are formed when two parallel lines are cut by a transversal and why the exterior angles of a triangle ...
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.