Cosine Rule

The correct formula for the Cosine Rule is $c^2=a^2+b^2-2ab\cdot \cos (C)$, where $a$ and $b$ are the sides of the triangle, and $C$ is the angle between them.

The Cosine Rule is used to find missing sides in a triangle when we know two sides and the included angle. It can also be used to find missing angles when we know all three sides of the triangle.

Using the Cosine Rule, we calculate $x^2=4^2+5^2-2\times 4\times 5\times \cos (60^{\circ })$. Since $\cos (60^{\circ })=0.5$, we have $x^2=16+25-2\times 20\times 0.5=21$. Therefore, $x=\sqrt{21}\text{ cm}$.

Using the Cosine Rule formula $x^2=a^2+b^2-2ab\cdot \cos (C)$, where $a=5$, $b=12$, and $C=60^{\circ }$, we calculate $x^2=5^2+12^2-2\times 5\times 12\times \cos (60^{\circ })$. Since $\cos (60^{\circ })=0.5$, we have $x^2=25+144-2\times 60\times 0.5=109$. Therefore, $x=\sqrt{109}\text{ cm}$.

Using the Cosine Rule formula $x^2=a^2+b^2-2ab\cdot \cos (C)$, where $a=6$, $b=10$, and $C=120^{\circ }$, we calculate $x^2=6^2+10^2-2\times 6\times 10\times \cos (120^{\circ })$. Since $\cos (120^{\circ })=-0.5$, we have $x^2=36+100-2\times 60\times (-0.5)=196$. Therefore, $x=\sqrt{196}=14\text{ cm}$.

Using the Cosine Rule: $c^2=a^2+b^2-2ab\cdot \cos (C)$, where $a=3$, $b=5$, and $c=7$. Substituting the values, we get $49=9+25-2\times 15\cdot \cos (C)$, which simplifies to $15=-30\cdot \cos (C)$. We find $\cos (C)=-\frac{1}{2}$, and using the inverse cosine function, we get $C=120^{\circ }$.