1. **State the problem:** We are given a triangle with sides and an angle, and the equation $$x^2 = 8.3^2 + c^2 - 2(8.3)(c) \cos 61^\circ$$ which comes from the cosine rule. We need to solve for $x$ in terms of $c$. 2. **Recall the cosine rule:** For any triangle with sides $a$, $b$, and $c$, and angle $A$ opposite side $a$, the cosine rule states: $$a^2 = b^2 + c^2 - 2bc \cos A$$ 3. **Identify the sides and angle:** Here, $x$ corresponds to side $a$, $8.3$ corresponds to side $b$, $c$ is side $c$, and the angle is $61^\circ$. 4. **Substitute the values into the formula:** $$x^2 = 8.3^2 + c^2 - 2 \times 8.3 \times c \times \cos 61^\circ$$ 5. **Calculate $8.3^2$ and $\cos 61^\circ$:** $$8.3^2 = 68.89$$ $$\cos 61^\circ \approx 0.4848$$ 6. **Rewrite the equation:** $$x^2 = 68.89 + c^2 - 2 \times 8.3 \times c \times 0.4848$$ 7. **Simplify the multiplication:** $$2 \times 8.3 = 16.6$$ $$16.6 \times 0.4848 \approx 8.048$$ So, $$x^2 = 68.89 + c^2 - 8.048c$$ 8. **Take the square root to solve for $x$:** $$x = \sqrt{68.89 + c^2 - 8.048c}$$ **Final answer:** $$x = \sqrt{68.89 + c^2 - 8.048c}$$ This expresses $x$ in terms of $c$ using the cosine rule.