Project Description

Write a C$++$ console application $($in Visual Studio$)$ that solves for the missing attributes of a triangle.

Imagine that you are an engineer in the field and have some of the measurements of a triangle, but not all. This application will solve the missing values.

e$.$g$.$ You have the length of two sides and the degrees of one of the angles. What is the length of the third side, and the degrees of the other two angles?

Solutio Triangulorum

Mathematicians have been triangle obsessed for millennia. The great mathematician Thales of Miletus century BCE$)$ solved the problem of measuring the height of the Great Pyramids without climbing them by using the congruence of triangles. Later, Euclid, Archimedes, and Eratosthenes revolutionized astronomy and engineering as they proved theorems that are equivalent to modern trigonometric formulae. By $140$BCE, Hipparchus published tables of ratios which became the first trigonometric tables.

A collection of those solutions can be found on Wikipedia. $[$h

Requirements

Write a C$++($Visual Studio $2022$ Solution$)$ program that reads an attribute string $($e$.$g$.,\text{'}$SSS$\text{'}$ for side$-$side$-$side, 'ASA' for angle$-$side$-$angle, etc.$)$ followed by three real numbers $($all on the same line$).$ Angles will be expressed in degrees $($e$.$g$.,87,$ where $180=$ radians$).$

Output to the console the length of the three sides $(abc)$ in one column, then the three opposite angles in the second column $().$ The angle $$ is the angle opposite line $a,$ etc. The angles must be reported in degrees.

Additionally, output:

the perimeter

the area

whether the triangle is equilateral, isosceles, or scalene.

whether the triangle is obtuse, right, or acute.

Align the numeric values in the columns.

Output all numeric values rounded to three decimal places.

Repeat the query until the user types "quit", "exit", or "bye" in any case.

$10$:$18$

Overview of particular steps and tools used when solving plane triangles

The standard method of solving the problem is to use fundamental relations.

Law of cosines

$a^2=b^2+c^2-2$bccos$$

$b^2=a^2+c^2-2$accos$$

$c^2=a^2+b^2-2$abcos$$

Law of sines

$\frac{a}{sin}=\frac{b}{sin}=\frac{c}{sin}$

Sum of angles

$++=180$

Law of tangents

$\frac{a-b}{a+b}=\frac{tan\frac{1}{2}(-)}{tan\frac{1}{2}(+)}$

There are other $($sometimes practically useful$)$ universal relations: the law of cotangents and Mollweide's formula.

Notes

To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle