The cosine rule
The sine rule always needs a side and its opposite angle. But what if you know two sides and the angle between them, and neither of them is opposite the quantity you are looking for? That is what the cosine rule is for: a generalised version of Pythagoras' theorem that works in any triangle.
A generalisation of Pythagoras
If the angle $C$ were $90^\circ$, then $\cos C=0$ and the formula becomes exactly Pythagoras' theorem. The cosine rule adds a correction term when the angle is not a right angle.
In a triangle $ABC$, sides $b=5$ cm and $c=8$ cm and the included angle $A=60^\circ$ are known. Calculate the length of side $a$.
In a triangle $ABC$, sides $b=5$ cm and $c=8$ cm and the included angle $A=60^\circ$ are known. Calculate the length of side $a$.
In a triangle $ABC$, the sides measure $a=10$ cm and $b=12$ cm and the included angle is $C=45^\circ$. Set up the cosine rule to calculate the exact value of side $c$.
In a triangle $ABC$, the sides measure $a=10$ cm and $b=12$ cm and the included angle is $C=45^\circ$. Set up the cosine rule to calculate the exact value of side $c$.
Finding a side: two sides and the included angle
If you know two sides and the angle between them, the third side comes straight out by substituting into the formula.
A parallelogram has two adjacent sides measuring 4 cm and 6 cm. If the included angle between them is $120^\circ$, calculate the length of the longer diagonal.
A parallelogram has two adjacent sides measuring 4 cm and 6 cm. If the included angle between them is $120^\circ$, calculate the length of the longer diagonal.
Calculate the distance between two inaccessible points $A$ and $B$, given that from an observation point $C$ the distances measured are $CA = 15$ m and $CB = 20$ m, and…
Calculate the distance between two inaccessible points $A$ and $B$, given that from an observation point $C$ the distances measured are $CA = 15$ m and $CB = 20$ m, and the visual angle $ACB$ is $40^\circ$.
Two aeroplanes depart from the same airport at the same time. Aeroplane A flies at $800$ km/h on a bearing of $040^\circ$ and aeroplane B flies at $650$ km/h on a…
Two aeroplanes depart from the same airport at the same time. Aeroplane A flies at $800$ km/h on a bearing of $040^\circ$ and aeroplane B flies at $650$ km/h on a bearing of $115^\circ$. What straight-line distance separates them after 1.5 hours of flight?
Finding an angle: all three sides
If you know all three sides, rearrange to find the cosine of the angle you want. The sign of the result tells you whether the angle is acute (positive cosine) or obtuse (negative cosine).
In a triangle $ABC$, the lengths of the sides are $a=3$ cm, $b=5$ cm, and $c=7$ cm. Calculate the size of angle $C$.
In a triangle $ABC$, the lengths of the sides are $a=3$ cm, $b=5$ cm, and $c=7$ cm. Calculate the size of angle $C$.
Determine the measure of angle $B$ in a triangle whose sides are $a=7$ cm, $b=8$ cm, and $c=9$ cm.
Determine the measure of angle $B$ in a triangle whose sides are $a=7$ cm, $b=8$ cm, and $c=9$ cm.
Determine the largest angle of a triangle whose sides measure 4 cm, 5 cm, and 6 cm.
Determine the largest angle of a triangle whose sides measure 4 cm, 5 cm, and 6 cm.
Inaccessible distances and forces
The cosine rule is the tool to use when two distances and the angle between them can be measured, but the distance you want cannot: surveying, navigation or combining forces.
To measure the distance across a lake between points $A$ and $B$, a surveyor stands at point $C$. He measures the distances $CA = 125,5$ m and $CB = 180,2$ m, and the…
To measure the distance across a lake between points $A$ and $B$, a surveyor stands at point $C$. He measures the distances $CA = 125,5$ m and $CB = 180,2$ m, and the angle $ACB = 75^\circ$. Calculate the distance $AB$.
A triangular roof truss has sides of 5 m, 7 m, and 9 m. Calculate the angle opposite the longest side.
A triangular roof truss has sides of 5 m, 7 m, and 9 m. Calculate the angle opposite the longest side.
In a triangle, the lengths of the sides are $x$, $x+2$, and $x+4$. If the largest angle measures $120^\circ$, set up the corresponding equation, determine the value of…
In a triangle, the lengths of the sides are $x$, $x+2$, and $x+4$. If the largest angle measures $120^\circ$, set up the corresponding equation, determine the value of $x$ and the exact lengths of the three sides.