Cosine Rule

Feedback: Not quite! 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.

Feedback: Not quite! 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.

Feedback: Not quite! 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, $x = \sqrt{21}$ cm.

Feedback: Not quite! 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, $x = \sqrt{109}$ cm.

Feedback: Not quite! 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, $x = \sqrt{196} = 14$ cm.

Feedback: Not quite! 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 $C = 120^\circ$.