1. **State the problem:** Calculate the value of $c$ using the formula $$c^2 = 59^2 + 317^2 - 2(59)(317)\cos 83^\circ$$ 2. **Formula used:** This is the Law of Cosines formula, which relates the lengths of sides of a triangle to the cosine of one of its angles. 3. **Calculate each term:** - $59^2 = 3481$ - $317^2 = 100489$ - Calculate $2 \times 59 \times 317 = 37306$ - Calculate $\cos 83^\circ \approx 0.12187$ 4. **Substitute values:** $$c^2 = 3481 + 100489 - 37306 \times 0.12187$$ 5. **Calculate the product:** $$37306 \times 0.12187 \approx 4546.3$$ 6. **Simplify:** $$c^2 = 3481 + 100489 - 4546.3 = 103970.7$$ 7. **Find $c$ by taking the square root:** $$c = \sqrt{103970.7} \approx 322.4$$ **Final answer:** $c \approx 322.4$