So we use the sinusoidal rule to find unknown lengths or angles of the triangle. It is also known as the sinusoidal rule, sinusoidal law or sinusoidal formula. When looking for the unknown angle of a triangle, the formula of the sinusoidal distribution can be written as follows: The cosine equation: a2 = b2 + c2 – 2bccos(A) (version given on the formula sheet to find the missing page) cos (A) = (b2 + c2 – a2) / 2bc (use this version to find a missing angle) In general, the sine law is used, To solve the triangle when we know two angles and one side or two angles and one side closed. This means that the sine law can be used if we have ASA (angle-side-angle) or AAS (angle-angle-angle-side) criteria. The triple scalar product OA ⋅ (OB × OC) is the volume of the parallelepiped formed by the position vectors of the vertices of the spherical triangle OA, OB and OC. This volume is invariant to the specific coordinate system used to represent OA, OB, and OC. The value of the triple scalar product OA ⋅ (OB × OC) is the determinant 3 × 3 with OA, OB and OC as lines. With the z-axis along OA, the square of this determinant is The sinusoidal equation: a / sinA = b / sinB = c / sinC (use to find the missing page) sinA / a = sinB / b = sinC / c (use to find the missing angle) In hyperbolic geometry, if the curvature is −1, the law of sine The law of sine is used, to find the unknown angle or page. Since these are calculated slightly differently, we can rearrange the sinusoidal rule to fit the part of the triangle we are trying to find. Here are the two versions. The law of sine is used to find the angle or unknown side of an oblique triangle. The oblique triangle is defined as any triangle that is not a right triangle.

The sinusoidal law must operate with at least two angles and its respective lateral dimensions simultaneously. A purely algebraic proof can be constructed from the spherical cosine law. From the sin identity 2 A = 1 − cos 2 A {displaystyle sin ^{2}A=1-cos ^{2}A} and from the explicit expression for cos A {displaystyle cos A} of the spherical law of cosine The answers are almost the same! (They would be exactly the same if we used perfect precision). The sinusoidal ruler can be used to find an angle of 3 sides and an angle or side of 3 angles and one side. The cosine rule can find a side of 2 sides and the angle contained or an angle of 3 sides. The law of sine in constant curvature is as follows We can use the sinusoidal rule to calculate a missing angle or side in a triangle when we have information about an angle and the opposite side and another angle and the opposite side. This means that if we divide side a by the sine of ∠A, it is equal to dividing side b by the sine of ∠ B, and also equal to dividing side c by sine of ∠C (or) The sides of a triangle are in the same proportion to each other as the sins of their opposite angles. The law of sine is one of two trigonometric equations commonly used to find lengths and angles in scale triangles, the other being the law of cosine. For example, when calculating the value of a in Example 2, the following calculation error is made.

It is easy to see how for small spherical triangles, when the radius of the sphere is much larger than the sides of the triangle, this formula becomes the plane formula at the limit, since the law of sine is usually used to find the angle or unknown side of a triangle. This law can be used when certain combinations of measurements of a triangle are given. This affects the formula of the sinusoidal rule because it looks like this and not the right way in Example 5. By substitution of K = 0, K = 1 and K = −1, we obtain the Euclidean, spherical and hyperbolic cases of the sinus distribution described above. This is wrong because the sinus is a function, so we can`t do that. The law of sine is used to determine the unknown side of a triangle when two angles and sides are given. Let pK(r) be the circumference of a circle of radius r in a space of constant curvature K. Then pK(r) = 2π sinK r. Therefore, the law of sine can also be expressed as follows: The spherical law of sines deals with triangles on a sphere whose sides are arcs of large circles.

The sinusoidal and consine rules apply to all triangles, you don`t necessarily need a right angle! The uppercase letters A, B, C are used to designate the vertices (corners) of a triangle, and the lowercase letters, a, b, c, are used to designate the sides of a triangle. To use the sinusoidal rule, we need to have sides and angles facing each other. If we find a missing page, then we need to know two angles and one of the opposite sides, we can use this information to find the side facing the other angle.