How To Calculate An Angle With Cosine Rule

Cosine Rule Formula:

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

Angle θ:

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1. What is the Cosine Rule?

The cosine rule (also known as the law of cosines) relates the lengths of the sides of a triangle to the cosine of one of its angles. It's particularly useful for solving triangles when you know:

All three sides (SSS), or
Two sides and the included angle (SAS)

2. How Does the Calculator Work?

The calculator uses the cosine rule formula:

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

Where:

\( a \) and \( b \) — The lengths of two sides of the triangle
\( c \) — The length of the side opposite the angle θ
\( \theta \) — The angle between sides a and b (in degrees)

Explanation: The formula calculates the cosine of the angle using the relationship between the sides, then takes the inverse cosine (arccos) to find the angle in radians, which is then converted to degrees.

3. When to Use the Cosine Rule

Details: The cosine rule is most useful when:

You know all three sides of a triangle (SSS) and need to find an angle
You know two sides and the included angle (SAS) and need to find the third side

It works for any type of triangle, not just right-angled triangles.

4. Using the Calculator

Tips:

Enter all three side lengths in the same units
Ensure the values satisfy the triangle inequality (sum of any two sides > third side)
The side labeled 'c' should be opposite the angle you want to calculate

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between cosine rule and sine rule?
A: The sine rule relates sides to opposite angles (a/sinA = b/sinB = c/sinC), while the cosine rule relates sides and included angles.

Q2: Can I use this for right-angled triangles?
A: Yes, but for right-angled triangles, basic trigonometry (SOHCAHTOA) is often simpler.

Q3: What if I get an error message?
A: Check that your side lengths satisfy the triangle inequality (sum of any two sides > third side).

Q4: How accurate are the results?
A: Results are accurate to 2 decimal places, but real-world accuracy depends on your input measurements.

Q5: Can I calculate a side length with this?
A: This calculator specifically finds angles. For side lengths, you would use the rearranged formula: c² = a² + b² - 2ab·cos(θ).