Solution of Triangles: Two Angles and Included Side are Given

When solving a triangle, we aim to find all the unknown sides and angles. In the case where two angles and the included side are given, we can determine the remaining parts of the triangle using trigonometric relationships and the Law of Sines.

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be written as:

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

where (a), (b), and (c) are the lengths of the sides opposite to angles (A), (B), and (C) respectively.

Given Information

For a triangle with given angles (A) and (B), and the included side (c), we can summarize the given information in the following table:

Given Angle	Given Side	Unknown Angle	Unknown Sides
(A)	(c)	(C)	(a), (b)

Steps to Solve the Triangle

1. Find the Third Angle: Since the sum of angles in a triangle is always 180 degrees, we can find the third angle (C) using the equation:

$$ C = 180^\circ - A - B $$

1. Use the Law of Sines: With angle (C) known, we can use the Law of Sines to find the lengths of the other two sides (a) and (b):

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

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

Solving these equations for (a) and (b) gives us:

$$ a = c \cdot \frac{\sin A}{\sin C} $$

$$ b = c \cdot \frac{\sin B}{\sin C} $$

Example

Let's consider a triangle with angles (A = 40^\circ), (B = 70^\circ), and the included side (c = 10) units.

1. Find the Third Angle:

$$ C = 180^\circ - 40^\circ - 70^\circ = 70^\circ $$

1. Use the Law of Sines:

$$ a = 10 \cdot \frac{\sin 40^\circ}{\sin 70^\circ} \approx 10 \cdot \frac{0.6428}{0.9397} \approx 6.84 \text{ units} $$

$$ b = 10 \cdot \frac{\sin 70^\circ}{\sin 70^\circ} = 10 \text{ units} $$

So, the lengths of the unknown sides are approximately (a \approx 6.84) units and (b = 10) units.

Important Points

• The sum of angles in any triangle is always 180 degrees.
• The Law of Sines can only be used when at least one angle-side opposite pair is known.
• If two angles are given, the third angle can be easily calculated.
• The Law of Sines can give two possible solutions for an angle (ambiguous case) when solving for angles, but this is not an issue when two angles are already known.
• Always check for the possibility of an obtuse angle, as the sine function has the same value for an angle and its supplement.

By following these steps and keeping these points in mind, one can solve any triangle given two angles and the included side.