Solution of Triangles: Two Sides and Included Angle Given

When solving a triangle, the objective is to find all the unknown sides and angles. In the case where two sides and the included angle are given, this is often referred to as the SAS (Side-Angle-Side) condition. This is one of the common scenarios in trigonometry and can be solved using the Law of Cosines.

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. If we have a triangle ABC, with sides a, b, and c opposite to angles A, B, and C respectively, the Law of Cosines states:

$$ c^2 = a^2 + b^2 - 2ab\cos(C) $$

$$ b^2 = a^2 + c^2 - 2ac\cos(B) $$

$$ a^2 = b^2 + c^2 - 2bc\cos(A) $$

Solving the Triangle

Given two sides, say a and b, and the included angle C, we can find the third side c using the Law of Cosines:

$$ c = \sqrt{a^2 + b^2 - 2ab\cos(C)} $$

Once we have all three sides, we can find the remaining angles using the Law of Sines or the Law of Cosines again.

Law of Sines

The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle:

$$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} $$

Using this law, we can find the other two angles:

$$ \sin(B) = \frac{b\sin(C)}{c} $$

$$ \sin(A) = \frac{a\sin(C)}{c} $$

Important Points and Differences

Point	Description
Law of Cosines	Used to find the third side of a triangle when two sides and the included angle are known.
Law of Sines	Used to find the unknown angles of a triangle once all sides are known.
SAS Condition	A condition where two sides and the included angle of a triangle are known, which is sufficient to solve the triangle.

Formulas

• Law of Cosines: $c = \sqrt{a^2 + b^2 - 2ab\cos(C)}$
• Law of Sines: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

Example

Let's solve a triangle where sides a = 5, b = 7, and the included angle C = 60°.

1. Find side c using the Law of Cosines:

$$ c = \sqrt{5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60°)} $$

$$ c = \sqrt{25 + 49 - 70 \cdot \frac{1}{2}} $$

$$ c = \sqrt{74 - 35} $$

$$ c = \sqrt{39} $$

1. Find angle B using the Law of Sines:

$$ \sin(B) = \frac{b\sin(C)}{c} $$

$$ \sin(B) = \frac{7\sin(60°)}{\sqrt{39}} $$

$$ \sin(B) = \frac{7\sqrt{3}/2}{\sqrt{39}} $$

$$ B = \sin^{-1}\left(\frac{7\sqrt{3}/2}{\sqrt{39}}\right) $$

1. Find angle A using the fact that the sum of angles in a triangle is 180°:

$$ A = 180° - B - C $$

After calculating B from the previous step, you can find A using the above equation.

This example demonstrates how to solve a triangle when two sides and the included angle are given. By applying the Law of Cosines and the Law of Sines, we can find all the unknown sides and angles.