Solving Acute, and Obtuse Triangles Using Law of Sines, and Law of Cosines.

Great, You learnt how to solve for right triangles using sine, cosine, or tangent! Now you have to solve for triangles that are not right, and you can not use sine, cosine, or tangent the way you had learnt before. What can you do now?

Law of Sines, and Law of cosines are the solution for this quandary. In the course of the lesson you will be introduced to each one of these laws, and then you will be given a set of examples. You may try to study the examples "A", and then go to solve the examples "B", and compare your answer with the solution presented in the lesson. You will find the lesson quite interesting indeed!

Lesson's Content

Lesson's Glossary

Acute angle
An angle whose measure is greater than 0 but less than 90 degrees.

Acute triangle
A triangle whose angles are acute.

Cosine
Ratio determined by the hypotenuse in a right triangle and a side adjacent to a reference angle.

Equilateral triangle
A triangle whose sides are equal in length.

Isosceles triangle
A triangle with two sides of equal length.

Proportion
A statement that two ratios are equal

Proportional
One of four numbers that form a true proportion

Sine
Ratio of the length of the opposite side to a reference angle in a right triangle and the hypotenuse. Opposite/hypotenuse.

Obtuse angle
An angle whose measure is greater than 90 but less than 180 degrees

Obtuse triangle
A triangle with one acute angle.

Ratio
A quotient of 2 numbers

Right triangle
A triangle that has a 90 degree angle.

Scalene triangle
A triangle with no equilateral sides.

Triangle
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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