Solving General Triangles

Solved Problems

Solution.

This is an example of the case, so first we find the unknown angle

Now using the Sine Rule, we can calculate the sides and

Solution.

We know two sides and the angle between them, that is, we have the case. Therefore, we will use the Law of Cosines to solve the triangle.

Find the side

The angle can also be found by the Cosine Rule:

We see that so is a right triangle.

The remaining angle is equal to

Solution.

We have here the $$ case. Find the angle $$ by the Cosine Rule:

Then

Similarly we calculate the angle $$

Hence,

The angle $$ is given by

Solution.

We are given an $$ triangle, which includes two sides and the angle opposite one of these sides. This set of parameters is classified as an ambiguous case since it may have more than one solution. The unique solution exists if only the angle is opposite the longest side of the two given sides. In our case $$ so the triangle is well defined.

Using the Law of Sines, we find the angle $$

It follows from here that

Determine the third angle $$

The side $$ can be found from the Law of Sines:

Solution.

Let the angle $$ be equal to $$ Write the Law of Cosines for triangle $$

and express $$ in terms of $$

Similarly, given that $$, write the Cosine Rule for $$

By the reduction identity, $$ Hence,

Substitute $$ from the previous equation to obtain the length of the longer diagonal $$

Hence,

Solution.

To prove the formula, we write the Sine Rule for the triangle:

It follows from this relationship that

Adding the two last equations gives us

Now we transform the expression in the numerator using the sum-to-product identity:

We also apply the double angle formula to $$ in the denominator. As a result, we get

Since $$ then using the cofunction identity, we have

that is,

Thus, we have proved Mollweide's formula