c. 2000 BC: Ancient Egyptians and Babylonians have developed understanding of the relationships between the sides of triangles. However, they do not know the concept of angle measures. This study is not quite trigonometry; instead, it is called trilaterometry. c. 1600 BC: The ancient Egyptians developed a very basic form of trigonometric operations. For example, the word seked was used to describe an angle's cotangent. This was the first recorded instance of true trigonometry. A Brief History of Trigonometry c. 200 BC: The ancient Greeks developed circle trigonometry. They inscribed triangles in circles to solve them, as the sides of the triangle would be chords of the circle. Similar to trigonometric relations, a table of chords was compiled, describing the angles between certain lengths of triangle sides. c. 900 AD: The Law of Sines is discovered by Persian mathematicians. THE LAW OF SINES Necessary skills: Identify the given information of a triangle

Decide how to solve for its other angles/sides

Apply the law of sines to the situation The Problem In some triangles, where 2 sides and an angle are given, where the angle is not between the two sides (SSA) there is more than one possible triangle to solve for. The black values represent given information. Why does this problem arise? It is because of the nature of the inverse sine operation In these certain cases, there are alwas 2 possibilities for the angle, where one possibility is the supplement of the other. We can also put this into a more geometric perspective. If the side opposite the given angle is shorter than the other given side, but longer than a hypothetical altitude drawn from the endpoint of the given side, then there are 2 possible angles. In other words, when a<b and a>bsinA A, B, and C are the angles of a triangle

while a, b, and c are the opposite sides

to their corresponding angles Using the Law of Sines, one can make an equation where the sine of the given angle over the given side equals the sine of the unknown angle over the sine of the other given side. However, when the equation is simplified, the result for the unknown angle is an arcsin expression. And in yet another perspective, it is when arcsin(b[sin(A)/a) has 2 solutions in the range [0,180] or [0,π] Trigonometry:

The Law of Sines Sample Problem: In a triangle ABC, given that angle A has a measure of 50 degrees and angle B has a measure of 65 degrees, while side a has a length of twelve, find the rest of the values of the triangle here is a link demonstrating the possibilities of the ambiguous case:

http://www.marckerschner.com/teaching-resources/law-of-sines-ambiguous-case/ This interactive animation helps give a visual idea of how the ambiguous case works, and under which conditions it appears.

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# Trigonometry

MI4 Project