8.1 Non-right Triangles: Law of Sines

Law of Sines and Its Applications

The Law of Sines lets you solve triangles that aren't right triangles by relating each side to the sine of its opposite angle. Since most real-world triangles aren't right triangles, this tool comes up constantly in applications like surveying, navigation, and engineering.

more resources to help you study

practice questions

Law of Sines for Non-Right Triangles

For any triangle with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$:

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

This ratio is constant across all three angle-side pairs in the triangle. It works for every triangle, not just right triangles. You can also flip it (sides on top) depending on whether you're solving for a side or an angle:

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

When can you use it? You need at least one angle-side opposite pair (an angle and the side across from it). That means the Law of Sines applies when you're given:

• AAS or ASA — two angles and one side
• SSA — two sides and an angle opposite one of them (the ambiguous case)

Note that AAA (three angles, no sides) does not let you solve for unique side lengths. Knowing all three angles only tells you the triangle's shape, not its size. You'd get infinitely many similar triangles.

Solving AAS / ASA Problems

1. Find the third angle using the fact that $A + B + C = 180°$.
2. Set up the Law of Sines ratio using the known angle-side pair.
3. Solve for each unknown side by cross-multiplying.

For example, if you know $A = 40°$, $B = 60°$, and $a = 10$:

• $C = 180° - 40° - 60° = 80°$
• $\frac{\sin 40°}{10} = \frac{\sin 60°}{b}$, so $b = \frac{10 \sin 60°}{\sin 40°} \approx 13.47$
• Repeat for $c$ using $\frac{\sin 40°}{10} = \frac{\sin 80°}{c}$

The Ambiguous Case (SSA)

SSA is the tricky one. When you're given two sides and an angle opposite one of them, there might be zero, one, or two valid triangles. Here's how to work through it:

1. Use the Law of Sines to find the sine of the unknown angle: $\sin B = \frac{b \sin A}{a}$.
2. Check the value of $\sin B$:
   • If $\sin B > 1$: no triangle exists (impossible sine value).
   • If $\sin B = 1$: one triangle with a right angle at $B$.
   • If $0 < \sin B < 1$: there are potentially two solutions, because $\sin B = \sin(180° - B)$. You need to check both:
     • $B_1 = \sin^{-1}(\sin B)$ (the acute angle)
     • $B_2 = 180° - B_1$ (the obtuse angle)
     • For each candidate, verify that $A + B < 180°$. If both work, you have two valid triangles. If only one works, you have one triangle.

The ambiguity arises because two different angles (one acute, one obtuse) can share the same sine value. Always check both possibilities.

Area Calculation with the Sine Function

When you know two sides and the included angle (the angle between them), you can find the area without needing the height:

$\text{Area} = \frac{1}{2}ab\sin C$

Here, $a$ and $b$ are the two known side lengths, and $C$ is the angle between them.

Steps:

1. Identify two sides and the angle between them (not opposite one of them).
2. Plug into the formula and compute.

For example, if $a = 8$, $b = 5$, and the included angle $C = 30°$:

$\text{Area} = \frac{1}{2}(8)(5)\sin 30° = \frac{1}{2}(8)(5)(0.5) = 10 \text{ square units}$

This formula is really just a generalization of $\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height}$, where $b\sin C$ gives you the height of the triangle relative to side $a$.

Real-World Applications of Law of Sines

For word problems, follow this general approach:

1. Sketch a triangle and label all given sides and angles.
2. Identify what you know: Do you have an angle-side opposite pair? Which case (AAS, ASA, SSA) applies?
3. Apply the Law of Sines to find the unknowns.
4. If the problem asks for area, use $\text{Area} = \frac{1}{2}ab\sin C$ once you have two sides and their included angle.
5. Interpret your answer in context with correct units (meters, feet, square meters, etc.).

Common application types include finding distances across rivers or lakes, determining heights of structures from two observation points, and calculating distances in navigation problems where bearings are given as angles.

Triangle Properties and Trigonometric Relationships

• Congruent triangles have the same shape and size (all corresponding sides and angles are equal).
• Similar triangles have the same shape but can differ in size (corresponding angles are equal, sides are proportional). AAA information produces similar triangles, which is why you can't find unique side lengths from angles alone.
• Trigonometric ratios (sine, cosine, tangent) are defined using right triangles. The Law of Sines extends the sine ratio to work in any triangle, which is what makes it so useful for non-right triangle problems.