Solve Abc 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

Results:

Enter known values to calculate the triangle

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1. What is the Law of Sines?

The Law of Sines is a fundamental relationship in trigonometry that relates the lengths of sides of a triangle to the sines of its opposite angles. The formula is:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]

where R is the radius of the triangle's circumcircle. This law applies to all triangles, not just right-angled ones.

2. How Does the Calculator Work?

The calculator uses the Law of Sines along with other trigonometric principles to solve triangles when given:

Two angles and one side (AAS or ASA)
Two sides and a non-included angle (SSA, which may have 0, 1, or 2 solutions)
Two sides and the included angle (SAS)
Three sides (SSS)

Note: The calculator automatically detects which case applies based on the provided information.

3. Types of Triangle Problems Solved

AAS/ASA: When two angles and any side are known, the calculator finds the remaining sides and angle using the Law of Sines.

SSA (Ambiguous Case): When two sides and a non-included angle are known, there may be two possible solutions, one solution, or no solution. The calculator checks all possibilities.

SAS: When two sides and the included angle are known, the calculator uses the Law of Cosines to find the third side, then the Law of Sines for the remaining angles.

SSS: When all three sides are known, the calculator uses the Law of Cosines to find the angles.

4. Using the Calculator

Instructions: Enter any three elements of the triangle (at least one side). The calculator will solve for the remaining elements. All angles should be in degrees.

Tips: For best results, provide either:

Two angles and one side
Two sides and one angle (including the case where the angle is between the two sides)
All three sides

5. Frequently Asked Questions (FAQ)

Q1: Why do I sometimes get two solutions?
A: In the SSA case (two sides and non-included angle), there can sometimes be two valid triangles that satisfy the given conditions. This is known as the ambiguous case.

Q2: What if I get "No solution exists"?
A: This means the given measurements cannot form a valid triangle. For SSA, it happens when sin(B) > 1. For SSS, it happens when the sum of any two sides isn't greater than the third.

Q3: Can I use this for right triangles?
A: Yes, the Law of Sines works for all triangles, including right triangles. However, right triangles can also be solved using simpler methods (Pythagorean theorem, basic trig ratios).

Q4: Why are angles limited to 180 degrees?
A: In Euclidean geometry, the sum of angles in a triangle is always exactly 180 degrees. No single angle can be 180° or more in a valid triangle.

Q5: How precise are the results?
A: Results are calculated with double precision but displayed with 2 decimal places. For exact values, symbolic computation would be needed.