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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.
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.
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 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.
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle .