Proving a triangle cannot exist with the cosine rule

Key takeaway: This proof combines the cosine rule with the range of the cosine function to test whether the given side lengths are possible.

Start with the cosine rule

For a triangle with sides $10$, $12$, and $26$, suppose the side opposite angle $C$ is $26$. The cosine rule gives

$

$26^2=10^2+12^2-2(10)(12)\cos C$

$

Substitute the values:

$

$676=100+144-240\cos C$

$

So

$

$676=244-240\cos C$

$

Rearrange:

$

$432=-240\cos C$

$

and therefore

$

$\cos C=-\frac{432}{240}=-1.8$

$

Check whether this is possible

The cosine of any real angle must lie between $-1$ and $1$. Since $-1.8$ is outside that range, no real triangle can satisfy these side lengths.

This is enough to prove the triangle is impossible.

Link to triangle inequality

The result also matches the triangle inequality idea: the sum of the two shorter sides must be greater than the longest side. But here

$

$10+12=22<26$

so the three lengths cannot form a triangle at all. The cosine-rule calculation confirms the same conclusion algebraically.

Final conclusion

It is impossible to have a triangle with sides measuring 10 cm, 12 cm, and 26 cm because the cosine rule produces an invalid cosine value.

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Pitfalls the pros know 👇 One trap is choosing the wrong side as the side opposite the angle in the cosine rule and then losing track of the algebra. Another is stopping once a negative cosine appears; negative values are fine, but values outside the interval $[-1,1]$ are not. A third mistake is ignoring the triangle inequality. If the longest side is already larger than the sum of the other two sides, the triangle cannot exist before you even apply trigonometry.

What if the problem changes? If the longest side were changed from 26 cm to 21 cm, the triangle inequality would no longer fail immediately because $10+12=22>21$. In that case, the cosine rule could be used to check whether the angle opposite the 21 cm side is valid. If the side lengths were 10 cm, 12 cm, and 22 cm exactly, then the triangle would be degenerate, with the two shorter sides lining up in a straight line. That is a different geometric case from a true triangle.

Tags: cosine rule, triangle inequality, invalid cosine value