Foundation – Unit 1
... Notes: Candidates should know not to round off values during the intermediate steps of a calculation. Candidates should be able to use a calculator to apply the four rules to fractions and decimals in problems. This is part of the core number work required across all units. The core number work will ...
... Notes: Candidates should know not to round off values during the intermediate steps of a calculation. Candidates should be able to use a calculator to apply the four rules to fractions and decimals in problems. This is part of the core number work required across all units. The core number work will ...
Here - UFL MAE
... c 2 a 2 b 2 2ab cos(C ) When C=/2 radians this just reduces to the Pythagorean Theorem. For an equilateral triangle we have cos(/3)=cos(60deg)=0.5. According to this law one also knows that one of the acute angles of a 3-4-5 right triangle equals =arcos(3/5)=53.130 deg. The other acute ang ...
... c 2 a 2 b 2 2ab cos(C ) When C=/2 radians this just reduces to the Pythagorean Theorem. For an equilateral triangle we have cos(/3)=cos(60deg)=0.5. According to this law one also knows that one of the acute angles of a 3-4-5 right triangle equals =arcos(3/5)=53.130 deg. The other acute ang ...
StewartPCalc6_06_05
... Suppose we are given two angles and the included side, then it’s clear that one and only one triangle can be formed (see Figure 2(a)). ...
... Suppose we are given two angles and the included side, then it’s clear that one and only one triangle can be formed (see Figure 2(a)). ...
File
... corresponding parts of the other triangle, the two right triangles are congruent If you know two sides of one right triangle are congruent to two sides of another right triangle, then you can prove the right triangles congruent no matter where the sides are ...
... corresponding parts of the other triangle, the two right triangles are congruent If you know two sides of one right triangle are congruent to two sides of another right triangle, then you can prove the right triangles congruent no matter where the sides are ...
EUCLIDEAN GEOMETRY Contents 1. Euclid`s geometry as a theory
... 2) If equals be added to equals, the wholes are equal. In other words, if a1 = a2 and b1 = b2 then a1 + b1 = a2 + b2 . This is true for numbers as well as for segments and angles. 3) If equals be subtracted from equals, the remainders are equal. In other words, if a1 = a2 and b1 = b2 then a1 − b1 = ...
... 2) If equals be added to equals, the wholes are equal. In other words, if a1 = a2 and b1 = b2 then a1 + b1 = a2 + b2 . This is true for numbers as well as for segments and angles. 3) If equals be subtracted from equals, the remainders are equal. In other words, if a1 = a2 and b1 = b2 then a1 − b1 = ...
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.