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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.
A counterargument can be used to rebut an objection to a premise, a main contention or a lemma. Synonyms of counterargument may include rebuttal, reply, counterstatement, counterreason, comeback and response. An attempt to rebut an argument may involve generating a counterargument, or finding a counterexample. [1]
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.
Definitions of objection vary in whether an objection is always an argument (or counterargument) or may include other moves such as questioning. [1] An objection to an objection is sometimes known as a rebuttal. [2] An objection can be issued against an argument retroactively from the point of reference of that argument.
In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems.
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. The proof proceeds by assuming that the opposite sides are not equal, and derives a contradiction.
Suppose the triangle is tilted even more until the water level on the right side is at 8 units. Predict what the water level in units will be on the left side. Typical Solutions. Someone with knowledge about the area of triangles might reason: "Initially the area of the water forming the triangle is 12 since 1 / 2 × 4 × 6 = 12. The ...
For example, that a genus was not predicable of the species, or that lines drawn from the centre to the circumference were not equal, or that a triangle did not have three angles equal to two right angles. [26] This can be done on a sphere, and not on a flat surface.