Cosine Rule: Complete Guide with Calculator & Examples

\[a^2 = b^2 + c^2 - 2bc \cos(A)\]

\[b^2 = a^2 + c^2 - 2ac \cos(B)\]

\[c^2 = a^2 + b^2 - 2ab \cos(C)\]

Result:

Result:

📏 Scenario 1: Find a Side (SAS)

Given: Two sides and the included angle
Find: The third side
Example: Sides = 7, 9; Angle between them = 60°
Use: \(a^2 = b^2 + c^2 - 2bc \cos(A)\)

📐 Scenario 2: Find an Angle (SSS)

Given: All three sides
Find: Any angle
Example: Sides = 5, 7, 9
Use: \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\)

\[a = \sqrt{67} \approx 8.19 \text{ cm}\]

\[A = \cos^{-1}(0.5238) \approx 58.41°\]

The cosine rule (also called the law of cosines) is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It states: \(a^2 = b^2 + c^2 - 2bc \cos(A)\), where \(a\), \(b\), \(c\) are the sides and \(A\) is the angle opposite side \(a\). It's used to find unknown sides or angles in non-right-angled triangles. The cosine rule is an extension of the Pythagorean theorem that works for all triangles, not just right-angled ones.

Use the cosine rule when: (1) You know two sides and the included angle (SAS) and need to find the third side, or (2) You know all three sides (SSS) and need to find an angle. Use the sine rule when: (1) You know two angles and one side (AAS/ASA), or (2) You know two sides and a non-included angle (SSA). The cosine rule is necessary for SAS and SSS cases because the sine rule doesn't work with these configurations. The cosine rule is also preferred when you want to avoid the ambiguous case that can occur with the sine rule.

To find angle \(A\), rearrange the cosine rule to: \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\), then calculate \(A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\) or \(A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\). This formula works when you know all three sides of the triangle. The rearrangement comes from solving the original formula for \(\cos(A)\) by moving terms around algebraically. Remember that the result of the fraction must be between -1 and 1 (the range of cosine values), otherwise you've made a calculation error or the sides don't form a valid triangle.

The sine rule relates sides to opposite angles using ratios: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\). The cosine rule relates all three sides to one angle: \(a^2 = b^2 + c^2 - 2bc \cos(A)\). The sine rule is simpler to use but only works in specific cases (when you know angle-side pairs), while the cosine rule works for SAS and SSS cases but requires more calculation (squaring, multiplication, square roots). The sine rule uses division and proportions, while the cosine rule uses quadratic relationships. Both are essential for solving all types of triangle problems in GCSE, IGCSE, A-Level, and IB Mathematics.

Yes! The cosine rule works for all triangles, including right-angled triangles. In fact, when you use the cosine rule on a right-angled triangle (where one angle is 90°), it simplifies to the Pythagorean theorem. Since \(\cos(90°) = 0\), the term \(-2bc \cos(A)\) becomes zero, leaving you with \(a^2 = b^2 + c^2\), which is Pythagoras' theorem. This shows that the cosine rule is actually a generalization of the Pythagorean theorem that works for any triangle. However, for right-angled triangles, it's usually easier to just use Pythagoras or basic trigonometry (SOH CAH TOA).

You can use either degrees or radians, but you must be consistent and ensure your calculator is in the correct mode. For GCSE and IGCSE Mathematics, angles are typically given in degrees, so make sure your calculator is in degree mode (DEG). For A-Level and IB Mathematics, you might work with radians in some contexts. The important thing is that when you press the cos button on your calculator, it interprets the angle correctly. Most calculator errors in the cosine rule come from being in the wrong angle mode. Always check your calculator mode before starting!

SAS stands for "Side-Angle-Side" – it means you know two sides of a triangle and the angle between them (the included angle). Use the cosine rule to find the third side. SSS stands for "Side-Side-Side" – it means you know all three sides of the triangle. Use the rearranged cosine rule to find any angle. Other abbreviations you might see include AAS (Angle-Angle-Side), ASA (Angle-Side-Angle), and SSA (Side-Side-Angle). These letters help you quickly identify what information you have and which formula to use for solving triangles.

Common reasons for errors: (1) Calculator in wrong mode – check it's in degree mode (DEG) if working in degrees, (2) Invalid triangle – the three sides don't satisfy the triangle inequality (the sum of any two sides must be greater than the third side), (3) Calculation error – double-check your arithmetic, especially when squaring and dividing, (4) Value outside cosine range – if your calculation gives a value greater than 1 or less than -1, the sides don't form a valid triangle or there's an arithmetic error. The inverse cosine function (\(\cos^{-1}\)) only accepts values between -1 and 1. Always verify your inputs form a valid triangle before applying the formula.