Law of Sines and Cosines Knowledge-Based Study

Study notes

1. Solving Non-Right Triangles

   In right-triangle trigonometry, basic trigonometric ratios like sine, cosine, and tangent are defined using the hypotenuse, adjacent, and opposite sides. However, many geometric problems involve oblique triangles—triangles that do not contain a right angle.

   Standard notation for any triangle $ABC$ uses uppercase letters $A$, $B$, and $C$ to denote the vertices and their corresponding interior angle measures. Lowercase letters $a$, $b$, and $c$ denote the lengths of the sides opposite those angles, respectively:

   • Side $a$ is opposite angle $A$.
   • Side $b$ is opposite angle $B$.
   • Side $c$ is opposite angle $C$.

   The Law of Sines and the Law of Cosines extend trigonometry to solve for unknown side lengths and angle measures in any arbitrary triangle, provided sufficient initial measurements are known.

   Statement and Derivation

   The Law of Sines establishes a direct proportionality between the lengths of the sides of a triangle and the sines of their opposite angles.

   For any triangle $ABC$:

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

   Equivalently, this relation can be written in reciprocal form:

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

   When to Apply the Law of Sines

   The Law of Sines is applicable whenever a known side and its opposite angle form a complete ratio, specifically in the following cases:

   • AAS (Angle-Angle-Side): Two angles and a non-included side are known.
   • ASA (Angle-Side-Angle): Two angles and the included side are known (the third angle is found using $A + B + C = 180^\circ$).
   • SSA (Side-Side-Angle): Two sides and a non-included angle are known (often referred to as the ambiguous case).

   Analyzing the SSA Configuration

   When given two sides and an angle opposite one of them (for example, $a$, $b$, and acute angle $A$), the given measurements may determine no triangle, exactly one triangle, or two distinct triangles. This occurs because $\sin \theta = \sin (180^\circ - \theta)$, meaning an acute angle and its obtuse supplement share the same sine value.

   Determining the Number of Solutions (for acute angle $A$)

   Let $h = b \sin A$ represent the altitude from vertex $C$ to side $c$:

   • Case 1 ($a < h$): Side $a$ is too short to reach the opposite side. No triangle exists.
   • Case 2 ($a = h$): Side $a$ forms a perpendicular segment to the base. Exactly one right triangle exists.
   • Case 3 ($h < a < b$): Side $a$ can swing to form two different triangles—one acute and one obtuse. Exactly two triangles exist.
   • Case 4 ($a \ge b$): Side $a$ can only form one valid triangle because the second possible position lies outside the boundary. Exactly one triangle exists.

   If angle $A$ is obtuse ($A \ge 90^\circ$), a valid triangle exists if and only if $a > b$; otherwise, no triangle can be formed.

   Statement and Forms

   The Law of Cosines generalizes the Pythagorean theorem to any triangle. When the included angle is $90^\circ$, $\cos 90^\circ = 0$, reducing the formula directly to $a^2 + b^2 = c^2$.

   For any triangle $ABC$:

   $$a^2 = b^2 + c^2 - 2bc \cos A$$$$b^2 = a^2 + c^2 - 2ac \cos B$$$$c^2 = a^2 + b^2 - 2ab \cos C$$

   Solving for Angles

   When all three sides are known, the equations can be rearranged to find the cosine of any interior angle:

   $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$$$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$$$$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$

   When to Apply the Law of Cosines

   • SAS (Side-Angle-Side): Two sides and the included angle are known. The law calculates the remaining side.
   • SSS (Side-Side-Side): All three side lengths are known. The law calculates the angle measures directly.

   Choosing the Optimal Strategy

   To avoid ambiguity when solving general triangles, follow these established best practices:

   • For SAS: Use the Law of Cosines to find the third side. Next, use the Law of Sines to find the angle opposite the shorter of the remaining sides (which must be acute), avoiding the ambiguous case. Finally, use the triangle angle sum to find the third angle.
   • For SSS: Use the Law of Cosines to calculate the largest angle first (the angle opposite the longest side). Since cosine values are uniquely negative for obtuse angles and positive for acute angles on $(0^\circ, 180^\circ)$, this immediately identifies whether the triangle has an obtuse angle. The remaining two angles must be acute.

   Trigonometric Area of a Triangle

   Using the sine function to express the altitude of a triangle yields the sine area formula:

   $$\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ac \sin B$$

   When only the three side lengths are available, Heron's Formula provides the area without calculating angles first:

   $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$

   where $s = \frac{a + b + c}{2}$ is the semi-perimeter of the triangle.