Law of Sine Triangle Calculator

Law of Sines Formula:

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

Side a:

units

Angle A (degrees):

Side b:

units

Angle B (degrees):

Side c:

units

Angle C (degrees):

Results:

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1. What is the Law of Sines?

The Law of Sines is a trigonometric formula relating the lengths of sides of a triangle to 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 in a triangle.

2. How Does the Calculator Work?

The calculator uses the Law of Sines formula:

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

Where:

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

Explanation: The calculator takes any combination of 3 elements (at least one side) and uses the Law of Sines to calculate the remaining sides and angles.

3. When to Use the Law of Sines

Details: The Law of Sines is particularly useful for:

Solving AAS (Angle-Angle-Side) triangles
Solving ASA (Angle-Side-Angle) triangles
Solving SSA (Side-Side-Angle) triangles (though this may have two solutions)

4. Using the Calculator

Tips:

Enter any three known values (must include at least one side)
Angles should be in degrees (0-180)
Side lengths can be in any consistent units
For SSA cases, the calculator will return one possible solution

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between Law of Sines and Law of Cosines?
A: Law of Sines relates sides to opposite angles, while Law of Cosines relates all three sides to one angle. Use Law of Sines when you know two angles and one side, or two sides and a non-included angle.

Q2: Why does my SSA case sometimes have two solutions?
A: The ambiguous case (SSA) can have two solutions when the given side opposite the given angle is shorter than the other given side but longer than the altitude of the triangle.

Q3: Can I use this for right triangles?
A: Yes, but right triangles can be solved more simply using basic trigonometric ratios (sine, cosine, tangent).

Q4: What units should I use for side lengths?
A: Any consistent units can be used (cm, inches, etc.) as long as all side lengths are in the same units.

Q5: How accurate are the results?
A: Results are accurate to two decimal places. For precise calculations, exact values may be needed.