Interactive Triangle Calculator

Law of Sines:

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

Side a:

units

Side b:

units

Side c:

units

Angle A:

degrees

Angle B:

degrees

Angle C:

degrees

Calculation Results:

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1. What is the Law of Sines?

The Law of Sines is a fundamental relationship in trigonometry that connects the lengths of sides of a triangle with the sines of its opposite angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles of the triangle.

2. How Does the Calculator Work?

The calculator uses the Law of Sines:

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

Where:

\( a, b, c \) — Lengths of the sides of the triangle
\( A, B, C \) — Angles opposite to sides a, b, c respectively

Explanation: Given any three elements of a triangle (including at least one side), the calculator can determine the remaining elements using trigonometric relationships.

3. Triangle Solving Basics

Details: To solve a triangle completely, you typically need to know:

Three sides (SSS)
Two sides and the included angle (SAS)
Two angles and one side (AAS or ASA)

The SSA case (two sides and a non-included angle) may have two, one, or no solutions.

4. Using the Calculator

Tips: Enter any three known values (including at least one side). The calculator will determine the remaining values. Angles should be in degrees, sides in any consistent units.

5. Frequently Asked Questions (FAQ)

Q1: What is the ambiguous case in triangle solving?
A: The SSA case can be ambiguous because it may produce two different triangles, one triangle, or no valid triangle depending on the values.

Q2: Can I use this calculator for right triangles?
A: Yes, though right triangles can also be solved using simpler Pythagorean theorem and basic trigonometric ratios.

Q3: What units should I use for the sides?
A: Any consistent units can be used (cm, inches, etc.) as long as all side lengths are in the same units.

Q4: How precise are the results?
A: Results are rounded to two decimal places. For exact values, symbolic computation would be needed.

Q5: What if I get an error message?
A: Make sure you've entered exactly three elements including at least one side, and that the values could form a valid triangle (e.g., angles sum to 180°).