The cosine rule is an essential trigonometric relationship used to solve oblique (non-right-angled) triangles when the sine rule cannot be applied directly. It also serves as a generalisation of the Pythagoras Theorem for any triangle.
See how a non-right-angled triangle is split into two right-angled triangles to derive the formula, and how SAS and SSS configurations are used.
For any triangle \(ABC\), side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
SAS: Two sides and the included angle are known.SSS: All three sides are known and an angle is needed.
Standard formula for side \(a\)
The isolated side matches the angle inside the cosine function.
Rearranged formula for angle \(A\)
Use this when all three side lengths are known.
Use the cosine rule to find a missing side or angle in any triangle.
\(a^2 = b^2 + c^2 - 2bc \cos A\)\(b^2 = a^2 + c^2 - 2ac \cos B\)\(c^2 = a^2 + b^2 - 2ab \cos C\)
\(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\)\(\cos B = \dfrac{a^2 + c^2 - b^2}{2ac}\)\(\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}\)
Drop an altitude \(h\) from vertex \(A\) to base \(BC\) at point \(D\). Let \(BD = x\), so \(DC = a - x\). Apply Pythagoras Theorem to both right-angled triangles and substitute \(x = c \cos B\).
If \(B = 90^\circ\), then \(\cos 90^\circ = 0\), so \(b^2 = a^2 + c^2\). The Pythagoras Theorem is a special case of the cosine rule.
When \(90^\circ < \theta < 180^\circ\), \(\cos \theta < 0\). Subtracting a negative term turns into addition, so \(a^2 > b^2 + c^2\).
For a side: square the two known sides, add them, subtract twice their product multiplied by the cosine of the included angle, then take the square root. For an angle: isolate the cosine, then use \(\cos^{-1}\).
Drop altitude \(h\) from \(A\) to \(BC\) at \(D\). Let \(BD = x\), so \(DC = a - x\).
In \(\triangle ACD\): \(b^2 = h^2 + (a - x)^2 = h^2 + a^2 - 2ax + x^2\).
In \(\triangle ABD\): \(c^2 = h^2 + x^2\), so \(h^2 = c^2 - x^2\).
Substitute \(h^2\) and \(x = c \cos B\) to obtain \(b^2 = a^2 + c^2 - 2ac \cos B\).
Cosine Rule for a side
Use for SAS condition.
Cosine Rule for an angle
Use for SSS condition.
Answer in your mind, then press "Check Answer".
In triangle \(ABC\), \(AC = 21\text{ cm}\), \(BC = 15\text{ cm}\), and \(\angle C = 35^\circ\). Find the length of \(AB\).
A triangle \(JKL\) has sides \(JK = 30\text{ cm}\), \(KL = 25\text{ cm}\), and \(JL = 35\text{ cm}\). Calculate the value of angle \(\angle KJL\).
In triangle \(PST\), \(PS = 6\text{ cm}\), \(ST = 6\text{ cm}\), and \(PT = 9\text{ cm}\). Find the value of \(\angle PST\).
The cosine rule can only be used on acute-angled triangles.
The cosine rule is universal and applies to acute, right-angled, and obtuse-angled triangles.
The cosine rule is completely separate from the Pythagoras Theorem.
The Pythagoras Theorem is a special case of the cosine rule when the included angle is \(90^\circ\) (\(\cos 90^\circ = 0\)).
You can use the cosine rule with any given combination of two sides and an angle.
To find the third side directly using the standard formula, the angle must be the included angle (the angle between the two known sides, SAS). If the angle is not included (SSA), the sine rule is typically used first.
Objective: Identify SAS/SSS conditions, obtuse angles, and key cosine rule components.
1 Under which of the following given conditions must the cosine rule be used instead of the sine rule to solve a triangle in the first step?
2 If \(\cos \theta = -0.1250\) when solving for an interior angle \(\theta\) of a triangle using the cosine rule, what type of angle is \(\theta\)?
3 For any triangle \(ABC\), the cosine rule formula to find side \(a\) is written as \(a^2 = b^2 + c^2 -\) _______ \(\cos A\).
4 When the included angle of a triangle is \(90^\circ\), the term \(2bc \cos 90^\circ\) becomes zero and the cosine rule simplifies to the _______ Theorem.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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