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  2. Special right triangle - Wikipedia

    en.wikipedia.org/wiki/Special_right_triangle

    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.

  3. File:30° 60° 90° Special Right Triangle.svg - Wikipedia

    en.wikipedia.org/wiki/File:30°_60°_90°_Special...

    The angle above the right angle is also represented by an arc, and its measure, 60°, is written below said arc. The hypotenuse (side opposite of the right angle) is of length 1, the side opposite the 30° angle is of length ½, and the side opposite the 60° angle is of length √3/2.

  4. Euler angles - Wikipedia

    en.wikipedia.org/wiki/Euler_angles

    The Haar measure for SO(3) in Euler angles is given by the Hopf angle parametrisation of SO(3), ⁡, [5] where (,) parametrise , the space of rotation axes. For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2 π , let z be uniform from −1 to 1, and let β = arccos( z ) .

  5. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    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.

  6. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two component rotations. He derived this formula in 1840 (see page 408). [3] The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation ...

  7. Right triangle - Wikipedia

    en.wikipedia.org/wiki/Right_triangle

    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).

  8. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    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. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes sin ⁡ ( 30 ∘ ) = 1 / 2 {\displaystyle \sin(30^{\circ ...

  9. Kepler triangle - Wikipedia

    en.wikipedia.org/wiki/Kepler_triangle

    [1] [8] [10] [11] Instead, the proportions of the pyramid can be adequately explained using integer ratios, based on a right triangle with sides 11 and 14. [1] [6] The name "Kepler triangle" for this shape was used by Roger Herz-Fischler, based on Kepler's 1597 letter, as early as 1979. [7]