Trigonometry: Law of Sines

The study of triangles often begins with right triangles, but what happens when no angle is 90 degrees? Oblique triangles—those without a right angle—are everywhere, from surveying land to analyzing forces in a bridge truss. To solve these triangles, you need tools that work beyond the Pythagorean theorem and basic SOHCAHTOA. The Law of Sines is your first and most powerful tool for unlocking the mysteries of oblique triangles, providing an elegant relationship between angles and their opposite sides that holds true in every single triangle, without exception.

The Fundamental Relationship

The Law of Sines establishes a constant ratio within any triangle. It states that the ratio of the length of a side to the sine of its opposite angle is the same for all three side-angle pairs. For a triangle with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, the law is written as:

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

This formula is powerful because it relates sides to angles across the entire triangle. A common way to think about it is that all three ratios are equal to the diameter of the triangle's circumscribed circle (the circle that passes through all three vertices). This relationship allows you to find missing sides or angles when you have a known pair (a side and its opposite angle) and one other piece of information. The proof is elegant: by dropping an altitude from one vertex, you create two right triangles, apply the definition of sine to each, and set the expressions for the altitude equal to each other, thereby deriving the proportional relationship.

Solving Triangles: AAS and ASA Cases

The most straightforward applications of the Law of Sines are the AAS (Angle-Angle-Side) and ASA (Angle-Side-Angle) cases. In both configurations, you know two angles and one side. Since the interior angles of any triangle sum to $180^{\circ }$, you can immediately find the third angle. At this point, you have a complete known side-angle pair and all three angles, allowing you to use the Law of Sines to find the two remaining sides.

Worked Example (AAS): Suppose you know angle $A=30^{\circ }$, angle $B=45^{\circ }$, and side $a=10$ (opposite angle $A$). First, find the third angle: $C=180^{\circ }-30^{\circ }-45^{\circ }=105^{\circ }$. Now, use the Law of Sines with the known pair $\frac{a}{\sin A}$ to find side $b$: $$\frac{10}{\sin 30^{\circ }}=\frac{b}{\sin 45^{\circ }}$$ Solving gives $b=\frac{10\cdot \sin 45^{\circ }}{\sin 30^{\circ }}=\frac{10\cdot (\sqrt{2}/2)}{1/2}=10\sqrt{2}\approx 14.14$. You would repeat the process with the same known ratio to solve for side $c$ using angle $C$.

The Ambiguous SSA Case

The SSA (Side-Side-Angle) case, where you know two sides and an angle not between them, is famously tricky and is known as the ambiguous case. Here, the given information may lead to zero, one, or two possible triangles. This ambiguity arises because the same values for side $a$, side $b$, and angle $A$ can sometimes be arranged in two different triangular configurations.

The key to resolving the ambiguity is to analyze the height of the potential triangle. Imagine you are given angle $A$ and adjacent side $b$, and you are tasked with finding side $a$ opposite angle $A$. The side $a$ must be long enough to reach the "base line" formed by the known elements. Let $h=b\sin A$ represent the height of this potential triangle.

• No Solution: If the given side $a$ is shorter than the height ($a<h$), it is too short to even reach the base line, forming no triangle.
• One Right Triangle Solution: If side $a$ is exactly equal to the height ($a=h$), it just reaches the base line, forming a single right triangle.
• Two Solutions: If side $a$ is *longer than the height but shorter than adjacent side $b\ast \ast \ast ($h < a < b$), it can swing to intersect the base line in two different places, creating two possible triangles (often called the acute and obtuse possibilities).
• One Solution: If side $a$ is *longer than or equal to side $b\ast \ast \ast ($a \ge b$), it is so long that it can only intersect the base line in one place, yielding a single triangle.

Worked Example (Ambiguous SSA): Given $A=40^{\circ }$, side $b=10$, and side $a=7$, determine the number of possible triangles.

1. Calculate the height: $h=b\sin A=10\sin 40^{\circ }\approx 6.43$.
2. Compare: $a=7$ and $h\approx 6.43$. Since $a>h$ but $a<b$ ($7<10$), we are in the "two solutions" scenario.

You would then use the Law of Sines to solve for both possible angle $B$ values: $\sin B=\frac{b\sin A}{a}$. The calculator gives $\sin B\approx 0.918$, leading to $B\approx 66.6^{\circ }$ (the acute solution) and its supplementary angle $B^{\prime }\approx 180^{\circ }-66.6^{\circ }=113.4^{\circ }$ (the obtuse solution). Both, when added to the given $A=40^{\circ }$, result in a sum less than $180^{\circ }$, so two valid triangles exist.

Common Pitfalls

1. Misapplying the Law for All Configurations: The Law of Sines is perfect for AAS, ASA, and SSA cases, but it is not the optimal first choice for the SAS (Side-Angle-Side) or SSS (Side-Side-Side) cases. For those, the Law of Cosines is a more direct tool. Attempting SSA with the Law of Cosines first often leads to a more complicated algebraic process.
2. Forgetting the Ambiguous Case in SSA: The most frequent critical error is blindly applying the Law of Sines to an SSA setup and accepting the first angle your calculator gives you via the ${\sin }^{-1}$ function. This function only returns angles between $-90^{\circ }$ and $90^{\circ }$ (or $0^{\circ }$ and $90^{\circ }$ for most triangle contexts), which is the acute solution. You must check if the supplementary angle ($180^{\circ }-\text{angle}$) is also a valid possibility by ensuring the sum of all angles would not exceed $180^{\circ }$.
3. Rounding Too Early in Multi-Step Problems: When using the Law of Sines sequentially to find multiple unknowns, using a rounded intermediate value (like an angle you just calculated) as input for the next step introduces propagated error. Always use the most precise value stored in your calculator from the previous step to maintain accuracy in your final answers.
4. Ignoring the Reality Check: After solving any triangle, perform a quick sanity check. Do the largest side and largest angle oppose each other? Do the three angles sum to approximately $180^{\circ }$ (allowing for minor rounding)? These simple checks can catch fundamental calculation errors.

Summary

• The Law of Sines ($\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$) provides a proportional relationship between all sides and their opposite angles in any triangle, making it essential for solving oblique triangles.
• It is the primary method for solving triangles in the AAS and ASA configurations, where you can find the third angle and then apply the law directly.
• The SSA (Side-Side-Angle) configuration is known as the ambiguous case because it can yield zero, one, or two possible triangular solutions, depending on the relative lengths of the given sides and the calculated height.
• To resolve the ambiguous case, compare the length of the side opposite the given angle ($a$) to the height $h=b\sin A$. The conditions $a<h$, $a=h$, $h<a<b$, and $a\ge b$ dictate the number of solutions.
• Always remember to check for the second possible (obtuse) angle in an ambiguous SSA case by considering the supplementary angle of the initial ${\sin }^{-1}$ result, provided the angle sum remains valid.