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For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
The trace of a rotation matrix is equal to the sum of its eigenvalues. For n = 2, a rotation by angle θ has trace 2 cos θ. For n = 3, a rotation around any axis by angle θ has trace 1 + 2 cos θ. For n = 4, and the trace is 2(cos θ + cos φ), which becomes 4 cos θ for an isoclinic rotation.
English: This file illustrates the special right triangle of angles 30°, 60° and 90°. A black square represents the borders of the file. Inside, the triangle is depicted with all of its special angles. The right angle is symbolized by a small square, and its measure, 90°, is written to the right and above it.
Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations , which have no fixed points, and (hyperplane) reflections , each of them having an entire ( n − 1) -dimensional flat of ...
x′-y′-z′ or x 1-y 1-z 1 (after first rotation) x″-y″-z″ or x 2-y 2-z 2 (after second rotation) X-Y-Z or x 3-y 3-z 3 (final) For the above-listed sequence of rotations, the line of nodes N can be simply defined as the orientation of X after the first elemental rotation. Hence, N can be simply denoted x′.
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
The right angle is marked as a small square. To the right of the 90° angle is another angle represented by an arc, and its measure, 45°, is written to the left of it. Atop the right angle is another angle symbolized by an arc, and its measure, 45°, is written below it. The hypotenuse (the side opposite the right angle) is of length 1.
The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.