Triangle Side Angles Calculator

Law of Sines:

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

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 that relates 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 (in degrees)

Explanation: The calculator can determine missing sides or angles when you know:

Two angles and any side (AAS or ASA cases)
Two sides and an opposite angle (SSA case)

3. When to Use the Law of Sines

Details: The Law of Sines is particularly useful for solving oblique triangles (non-right triangles). It's most effective when you know either:

Two angles and one side (AAS or ASA)
Two sides and one opposite angle (SSA)

For cases with two sides and the included angle (SAS) or three sides (SSS), the Law of Cosines is more appropriate.

4. Using the Calculator

Tips: Enter any three elements of the triangle (at least one side). The calculator will solve for the remaining elements. Note that in the SSA case (two sides and a non-included angle), there may be two solutions, one solution, or no solution.

5. Frequently Asked Questions (FAQ)

Q1: What if I get an error message?
A: This means the values you entered don't form a valid triangle. Check that angles sum to less than 180° and sides follow triangle inequality rules.

Q2: Why does the calculator sometimes give two solutions?
A: In the SSA case (ambiguous case), there can be two different triangles that satisfy the given conditions.

Q3: Can I use this for right triangles?
A: Yes, but right triangles can be solved more simply with Pythagorean theorem and basic trig functions.

Q4: What units should I use?
A: The units for sides can be anything (cm, inches, etc.) as long as they're consistent. Angles must be in degrees.

Q5: How accurate are the results?
A: Results are accurate to 4 decimal places. For exact values, use exact trigonometric ratios when possible.