Solve For Angle Calculator

Law of Cosines:

\[ \theta = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]

Angle θ:

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1. What is the Law of Cosines?

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful for solving triangles when you know all three sides or two sides and the included angle.

2. How Does the Calculator Work?

The calculator uses the Law of Cosines formula:

\[ \theta = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]

Where:

\( \theta \) — Angle opposite to side a (in radians)
\( a \) — Length of side opposite to angle θ
\( b, c \) — Lengths of the other two sides
\( \arccos \) — Inverse cosine function

Explanation: This formula calculates the angle θ when all three sides of the triangle are known, using the inverse cosine function to find the angle from the cosine ratio.

3. Importance of Angle Calculation

Details: Calculating angles in triangles is essential in various fields including engineering, architecture, navigation, and computer graphics. The Law of Cosines provides a reliable method for determining unknown angles when side lengths are known.

4. Using the Calculator

Tips: Enter all three side lengths in the same units. Ensure the values satisfy the triangle inequality theorem (sum of any two sides must be greater than the third side). The calculator will return the angle in degrees.

5. Frequently Asked Questions (FAQ)

Q1: What is the triangle inequality theorem?
A: The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Q2: When should I use Law of Cosines vs Law of Sines?
A: Use Law of Cosines when you know all three sides (SSS) or two sides and the included angle (SAS). Use Law of Sines when you know two angles and one side (AAS or ASA).

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

Q4: Why does the calculator sometimes show an error?
A: Errors occur when the input side lengths cannot form a valid triangle (violate triangle inequality) or when the cosine value falls outside the valid range of -1 to 1.

Q5: Can this calculator find other angles in the triangle?
A: This calculator finds the angle opposite to side a. To find other angles, rearrange the sides accordingly or use the calculated angle with the triangle sum theorem (angles sum to 180°).