1. **State the problem:** We need to find the magnitudes of two forces, say $F_1$ and $F_2$, given that their resultant is $R_1$ when they act at right angles (90°), and $R_2$ when they act at an angle of 60°. 2. **Write down the given information:** - Resultant at 90°: $R_1$ - Resultant at 60°: $R_2$ 3. **Use the formula for resultant of two forces:** - When forces act at angle $\theta$, the resultant $R$ is given by: $$ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} $$ 4. **Apply the formula for $\theta = 90^\circ$:** Since $\cos 90^\circ = 0$, $$ R_1 = \sqrt{F_1^2 + F_2^2} $$ Squaring both sides: $$ R_1^2 = F_1^2 + F_2^2 $$ 5. **Apply the formula for $\theta = 60^\circ$:** Since $\cos 60^\circ = \frac{1}{2}$, $$ R_2 = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \times \frac{1}{2}} = \sqrt{F_1^2 + F_2^2 + F_1 F_2} $$ Squaring both sides: $$ R_2^2 = F_1^2 + F_2^2 + F_1 F_2 $$ 6. **Substitute $F_1^2 + F_2^2$ from step 4 into step 5:** $$ R_2^2 = R_1^2 + F_1 F_2 $$ Rearranged: $$ F_1 F_2 = R_2^2 - R_1^2 $$ 7. **We now have two equations:** $$ F_1^2 + F_2^2 = R_1^2 $$ $$ F_1 F_2 = R_2^2 - R_1^2 $$ 8. **Let $x = F_1$ and $y = F_2$.** From the first equation: $$ x^2 + y^2 = R_1^2 $$ From the second: $$ xy = R_2^2 - R_1^2 $$ 9. **Find $x$ and $y$ by solving the quadratic:** Note that: $$ (x - y)^2 = x^2 - 2xy + y^2 = (x^2 + y^2) - 2xy = R_1^2 - 2(R_2^2 - R_1^2) = 3 R_1^2 - 2 R_2^2 $$ 10. **Calculate $x - y$:** $$ x - y = \sqrt{3 R_1^2 - 2 R_2^2} $$ 11. **Use the system:** $$ x + y = S \quad \text{(unknown sum)} $$ $$ x - y = D = \sqrt{3 R_1^2 - 2 R_2^2} $$ 12. **Express $x$ and $y$ in terms of $S$ and $D$:** $$ x = \frac{S + D}{2}, \quad y = \frac{S - D}{2} $$ 13. **Use $xy = R_2^2 - R_1^2$ to find $S$:** $$ xy = \frac{S^2 - D^2}{4} = R_2^2 - R_1^2 $$ $$ S^2 = 4(R_2^2 - R_1^2) + D^2 = 4(R_2^2 - R_1^2) + 3 R_1^2 - 2 R_2^2 = 2 R_2^2 + R_1^2 $$ 14. **Therefore:** $$ S = \sqrt{2 R_2^2 + R_1^2} $$ 15. **Final magnitudes:** $$ F_1 = \frac{S + D}{2} = \frac{\sqrt{2 R_2^2 + R_1^2} + \sqrt{3 R_1^2 - 2 R_2^2}}{2} $$ $$ F_2 = \frac{S - D}{2} = \frac{\sqrt{2 R_2^2 + R_1^2} - \sqrt{3 R_1^2 - 2 R_2^2}}{2} $$ **Answer:** The magnitudes of the two forces are given by the above expressions in terms of $R_1$ and $R_2$.