Sine & Cosine Laws Calculator: Step-by-Step

Q: What is the Ambiguous Case in the Law of Sines?

The ambiguous case occurs in an SSA scenario. Because sine is positive in two quadrants, it might yield two valid triangles, one valid triangle, or zero.

Q: When should I use the Law of Cosines instead of Sines?

Use the Law of Cosines for SAS or SSS scenarios, where you do not have a complete pair of a known side and its corresponding opposite angle.

Q: Do these laws work on right triangles?

Yes. If you plug a 90-degree angle into the Law of Cosines, the cos(90) part becomes 0, leaving exactly the Pythagorean theorem.

Navigating Non-Right Triangles

While the Pythagorean Theorem is perfect for right-angled triangles, the vast majority of triangles in mathematics and real-world engineering are "oblique" (meaning they do not contain a 90-degree angle). To solve for missing sides and angles in these shapes, mathematicians rely on two highly powerful trigonometric properties: The Law of Sines and The Law of Cosines.

Graph showing the area under a curve representing a definite integral

Before applying these laws, it is crucial to understand the standard geometric naming convention: capital letters (\( A, B, C \)) always represent the interior angles (usually measured in degrees), while lowercase letters (\( a, b, c \)) represent the lengths of the sides directly opposite those corresponding angles.

The Law of Sines

The Law of Sines establishes that the ratio of a side's length to the sine of its opposite angle is constant for all three sides of the triangle. It is written as:

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

You should deploy the Law of Sines when you are provided with either two angles and one side (AAS or ASA), or two sides and an angle strictly opposite one of them (SSA - the ambiguous case). By setting up a simple cross-multiplication algebra equation, you can quickly isolate the unknown variable.

The Law of Cosines

The Law of Cosines is essentially an expanded, more robust version of the Pythagorean theorem, adapted to account for the angle that deviates from 90 degrees. It is written as:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

This formula is your primary tool when you are given two sides and the included angle directly between them (SAS), or when you are given all three sides (SSS) and need to calculate the interior angles. It algebraically guarantees a single, definitive shape.

Frequently Asked Questions

What is the "Ambiguous Case" in the Law of Sines? +

The ambiguous case occurs in an SSA (Side-Side-Angle) scenario. Because the sine function is positive in both the first and second quadrants, solving for an angle using the inverse sine might yield two valid mathematical triangles, one valid triangle, or zero possible triangles depending on the side lengths.

When should I use the Law of Cosines instead of Sines? +

You must use the Law of Cosines when you do not have a complete "pair" (a known side and its known opposite angle). If you only know Side-Angle-Side (SAS) or Side-Side-Side (SSS), the Law of Sines will not work because the algebraic equation will contain two unknown variables.

Do these laws work on right triangles? +

Yes, they work perfectly on right triangles. In fact, if you plug a 90-degree angle into the Law of Cosines, \( \cos(90^\circ) \) equals 0. The entire end of the equation (\( -2ab \cdot \cos(C) \)) cancels out, leaving you with exactly the Pythagorean theorem: \( c^2 = a^2 + b^2 \).