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Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
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.
The shaded blue and green triangles, and the red-outlined triangle are all right-angled and similar, and all contain the angle . The hypotenuse B D ¯ {\displaystyle {\overline {BD}}} of the red-outlined triangle has length 2 sin θ {\displaystyle 2\sin \theta } , so its side D E ¯ {\displaystyle {\overline {DE}}} has length 2 sin 2 θ ...
Given triangle sides b and c and angle γ there are sometimes two solutions for a. The theorem is used in solution of triangles , i.e., to find (see Figure 3): the third side of a triangle if two sides and the angle between them is known: c = a 2 + b 2 − 2 a b cos γ ; {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}\,;}
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [1] [2] A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel in 1746. Thomas Simpson published the now-standard expression in 1748.
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
In this right triangle, denoting the measure of angle BAC as A: sin A = a / c ; cos A = b / c ; tan A = a / b . Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labeled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point.
Using the usual notations for a triangle (see the figure at the upper right), where a, b, c are the lengths of the three sides, A, B, C are the vertices opposite those three respective sides, α, β, γ are the corresponding angles at those vertices, s is the semiperimeter, that is, s = a + b + c / 2 , and r is the radius of the inscribed circle, the law of cotangents states that
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related to: math formulas for triangles examples with solutions answer page 6 8 9