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  2. Law of cosines - Wikipedia

    en.wikipedia.org/wiki/Law_of_cosines

    The third formula shown is the result of solving for a in the quadratic equation a 2 − 2ab cos γ + b 2c 2 = 0. This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data.

  3. Law of sines - Wikipedia

    en.wikipedia.org/wiki/Law_of_sines

    In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, ⁡ = ⁡ = ⁡ =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.

  4. 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.

  5. Proofs of trigonometric identities - Wikipedia

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

    This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...

  6. Pythagorean trigonometric identity - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_trigonometric...

    Point P has a positive y-coordinate, and sin θ = sin(π−θ) > 0. As θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant.

  7. Spherical law of cosines - Wikipedia

    en.wikipedia.org/wiki/Spherical_law_of_cosines

    As a special case, for C = ⁠ π / 2 ⁠, then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem: cosc = cos ⁡ a cosb {\displaystyle \cos c=\cos a\cos b\,} If the law of cosines is used to solve for c , the necessity of inverting the cosine magnifies rounding errors when c is small.

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  9. Solution of triangles - Wikipedia

    en.wikipedia.org/wiki/Solution_of_triangles

    If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ ′ = 180° − γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C ′, side b ′ and the angle γ ′ as the second solution. Once γ is obtained, the third angle α = 180° − β − γ.