Let's **solve the system step-by-step** by simplifying both equations. --- ## **Step 1: Write the Equations Clearly** Let’s denote \( a_3 \) and \( a_4 \) as unknowns. ### **Equation 1:** \[ -.12 \times 55.88^2 (\cos 60^\circ) - (a_3 \sin 20.066^\circ - 1 \times -1.6641)^2 (\cos 20.066^\circ) + (-.4 a_4 \cos 71.13^\circ) + (.4 \times 13.83418^2 \cos 71.13^\circ) = ? \] But from the image, the terms are: \[ (-.12 \times 55.88^2 (\cos 60^\circ)) - (1 \times a_3 (\sin 20.066^\circ) - 1 \times (-1.6641))^2 (\cos 20.066^\circ) + (-.4 \times a_4 (\cos 71.13^\circ)) + (.4 \times 13.83418^2 (\cos 71.13^\circ)) \] ### **Equation 2:** \[ (-.12 \times (55.88)^2 (i \sin 60^\circ)) + (1 \times i a_3 (\cos 20.066^\circ) - 1 \times (-1.6641)^2 (i \sin 20.066^\circ) + (-.4 \times i a_4 (\cos 71.13^\circ)) + .4 \times 13.83418^2 (i \sin 71.13^\circ) \] --- ## **Step 2: Calculate Trigonometric Values and Constants** - \(\cos 60^\circ = .5\) - \(\sin 20.066^\circ \approx .3436\) - \(\cos 20.066^\circ \approx .9394\) - \(\sin 60^\circ \approx .866\) - \(\cos 71.13^\circ \approx .3251\) - \(\sin 71.13^\circ \approx .9457\) --- ## **Step 3: Plug in the Values** #### **Equation 1:** Calculate each term: - \(55.88^2 = 3124.1744\) - \(13.83418^2 = 191.461\) - \(-1.6641^2 = 2.7702\) So, \[ \text{First term:}\quad -.12 \times 3124.1744 \times .5 = -187.45 \] \[ \text{Second term:}\quad (a_3 \times .3436 - 1 \times -1.6641)^2 \times .9394 \] But the image shows: \[ - (a_3 \sin 20.066^\circ - (-1.6641))^2 (\cos 20.066^\circ) \] So: \[ a_3 \sin 20.066^\circ + 1.6641 \] So the squared term is: \[ (a_3 \times .3436 + 1.6641)^2 \times .9394 \] \[ \text{Third term:}\quad -.4 a_4 \times .3251 = -.13004 a_4 \] \[ \text{Fourth term:}\quad .4 \times 191.461 \times .3251 = 24.892 \] Combine: \[ -187.45 - (a_3 \times .3436 + 1.6641)^2 \times .9394 - .13004 a_4 + 24.892 = \] #### **Equation 2:** \[ -.12 \times 3124.1744 \times i \times .866 = -324.34i \] \[ i a_3 \times .9394 \] \[ -2.7702 \times i \times .3436 = -.952i \] \[ -.4 i a_4 \times .3251 = -.13004 i a_4 \] \[ .4 \times 191.461 \times i \times .9457 = 72.368i \] Combine: \[ -324.34i + .9394 i a_3 - .952i - .13004 i a_4 + 72.368i = \] --- ## **Step 4: Collect Like Terms** **Equation 1:** \[ -187.45 - .9394(a_3 \times .3436 + 1.6641)^2 - .13004 a_4 + 24.892 = \] **Equation 2 (divide by i):** \[ -324.34 + .9394 a_3 - .952 - .13004 a_4 + 72.368 = \] Combine constants: \[ -324.34 - .952 + 72.368 = -252.924 \] So, \[ -252.924 + .9394 a_3 - .13004 a_4 = \] --- ## **Step 5: Solve for \(a_3\) and \(a_4\)** **Equation 2:** \[ .9394 a_3 - .13004 a_4 = 252.924 \] Let’s call this (1). --- **Equation 1 (expand square):** Let \(x = a_3\): \[ (a_3 \times .3436 + 1.6641)^2 = (.3436 a_3 + 1.6641)^2 = .1181 a_3^2 + 2 \times .3436 \times 1.6641 a_3 + (1.6641)^2 \] \[ = .1181 a_3^2 + 1.1442 a_3 + 2.7702 \] Multiply by .9394: \[ .9394 \times .1181 = .1109 \] \[ .9394 \times 1.1442 = 1.075 \] \[ .9394 \times 2.7702 = 2.5998 \] So, \[ -.1109 a_3^2 - 1.075 a_3 - 2.5998 \] Now, Equation 1 becomes: \[ -187.45 - (.1109 a_3^2 + 1.075 a_3 + 2.5998) - .13004 a_4 + 24.892 = \] Combine constants: \[ -187.45 - 2.5998 + 24.892 = -165.1578 \] So, \[ -.1109 a_3^2 - 1.075 a_3 - .13004 a_4 - 165.1578 = \] or \[ .1109 a_3^2 + 1.075 a_3 + .13004 a_4 = -165.1578 \] Let’s call this (2). --- ## **Step 6: Solve the System** From (1): \[ .9394 a_3 - .13004 a_4 = 252.924 \] Solve for \(a_4\): \[ .13004 a_4 = .9394 a_3 - 252.924 \] \[ a_4 = \frac{.9394 a_3 - 252.924}{.13004} \] Plug into (2): \[ .1109 a_3^2 + 1.075 a_3 + .13004 a_4 = -165.1578 \] Substitute \(a_4\): \[ .1109 a_3^2 + 1.075 a_3 + .13004 \left(\frac{.9394 a_3 - 252.924}{.13004}\right) = -165.1578 \] \[ .1109 a_3^2 + 1.075 a_3 + .9394 a_3 - 252.924 = -165.1578 \] \[ .1109 a_3^2 + (1.075 + .9394) a_3 - 252.924 + 165.1578 = \] \[ .1109 a_3^2 + 2.0144 a_3 - 87.7662 = \] --- ## **Step 7: Quadratic Formula** \[ a_3 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, - \(a = .1109\) - \(b = 2.0144\) - \(c = -87.7662\) Calculate discriminant: \[ D = b^2 - 4ac = (2.0144)^2 - 4 \times .1109 \times (-87.7662) \] \[ = 4.058 \,+\, 38.953 \,=\, 43.011 \] So, \[ a_3 = \frac{-2.0144 \pm \sqrt{43.011}}{2 \times .1109} \] \[ \sqrt{43.011} \approx 6.557 \] So: \[ a_3 = \frac{-2.0144 \pm 6.557}{.2218} \] Calculate both solutions: 1. \(a_3 = \frac{4.5426}{.2218} \approx 20.48\) 2. \(a_3 = \frac{-8.5714}{.2218} \approx -38.65\) --- Now, use first equation to solve for \(a_4\): \[ a_4 = \frac{.9394 a_3 - 252.924}{.13004} \] For \(a_3 = 20.48\): \[ a_4 = \frac{.9394 \times 20.48 - 252.924}{.13004} = \frac{19.241 - 252.924}{.13004} = \frac{-233.683}{.13004} \approx -1797.1 \] For \(a_3 = -38.65\): \[ a_4 = \frac{.9394 \times (-38.65) - 252.924}{.13004} = \frac{-36.29 - 252.924}{.13004} = \frac{-289.214}{.13004} \approx -2224.4 \] --- ## **Step 8: Final Answers** ### **Summary Table** | Solution | \( a_3 \) | \( a_4 \) | |----------|:----------:|:------------:| | (1) | 20.48 | -1797.1 | | (2) | -38.65 | -2224.4 | --- ### **Boxed Answer:** \[ \boxed{ \begin{aligned} & a_3 \approx 20.48,\quad a_4 \approx -1797.1 \\[1em] & \text{or} \\[1em] & a_3 \approx -38.65,\quad a_4 \approx -2224.4 \end{aligned} } \] --- **Let me know if you need further breakdown or interpretation!**