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  2. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.

  3. Sum of angles of a triangle - Wikipedia

    en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle

    In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. It was unknown for a long time whether other geometries exist, for which this sum is different. The influence of this ...

  4. Internal and external angles - Wikipedia

    en.wikipedia.org/wiki/Internal_and_external_angles

    The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...

  5. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number , except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°).

  6. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or ⁠ π / 2 ⁠ radians. Therefore ⁡ and ⁡ represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.

  7. Spherical trigonometry - Wikipedia

    en.wikipedia.org/wiki/Spherical_trigonometry

    The angles of proper spherical triangles are (by convention) less than π, so that < + + < (Todhunter, [1] Art.22,32). In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians.

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  9. Trigonometry of a tetrahedron - Wikipedia

    en.wikipedia.org/wiki/Trigonometry_of_a_tetrahedron

    The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge. The 4 solid angles - associated to each point of the tetrahedron.

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