An angle whose measure is greater than 0 but less than 90 degrees.
A triangle whose angles are acute.
Ratio determined by the hypotenuse in a right triangle and a side adjacent to a reference angle.
A triangle whose sides are equal in length.
A triangle with two sides of equal length.
A statement that two ratios are equal
One of four numbers that form a true proportion
Ratio of the length of the opposite side to a reference angle in a right triangle and the hypotenuse. Opposite/hypotenuse.
An angle whose measure is greater than 90 but less than 180 degrees
A triangle with one acute angle.
A quotient of 2 numbers
A triangle that has a 90 degree angle.
A triangle with no equilateral sides.
A polygon with three sides.
Interactive Geometric Applets: Relevant Theorems.
The Ambiguous Case: The law of sines to solve triangles, has an ambiguous case;
where two distinct triangles can be constructed (We have two different possible solutions to the triangle).
Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:
a) The only information known about the triangle is the angle A and two sides a and b, for which the angle A is not
the included angle of the two sides, in other words the angle is opposite to one of the given sides.
b) The angle A is acute.
c) The side a (opposite to the angle) is shorter than the side b, or a < b.
d) The side a (opposite to the angle) is longer than the altitude of a right angled triangle with angle A
and hypotenuse b, or a > b sin A.
The applet below illustrates using Law of Sines for the ambiguous case.
Alternative solution to the ambiguous case:
Using both Law of Cosines, and Law of Sines. You will apply the Law of Cosines, and get a second
degree equation, from which applying the Quadratic Formula will give you
two solutions, that with the Law of Sines will allow you to find the remaining values.
Solving by Law of Sines, or Law of Cosines does not use sine, and cosine the way you
do it with right triangles; nevertheless, you still need to get the inverse of sine,
and cosine. The following applet helps you to understand the relationship
between the trigonometric ratios, and their inverses. Drag any vertex to
view in a dynamic way this relationship.
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