Determine the value of the actute angle \( x \) in each of the following equations. \( \begin{array}{ll}\text { (1) } \sin \left(x-20^{\circ}\right)=0,678 & \text { (2) } 3 \cos \left(x+30^{\circ}\right)=2,121 \\ \text { (3) } 2 \tan \left(2 x-10^{\circ}\right)=3,4641 & \text { (4) } 2 \tan 2 x-10=3,4641\end{array} \)

Let's solve each equation step by step to determine the acute angle \( x \) for each case. ### Equation (1) \[\sin (x - 20^\circ) = 0.678\] To find \( x \): 1. Use the inverse sine function: \[x - 20^\circ = \arcsin(0.678)\] 2. Calculate \( \arcsin(0.678) \): \[x - 20^\circ \approx 42.40^\circ\] 3. Solve for \( x \): \[x = 42.40^\circ + 20^\circ \approx 62.40^\circ\] ### Equation (2) \[3 \cos (x + 30^\circ) = 2.121\] First, isolate \(\cos\): \[ \cos(x + 30^\circ) = \frac{2.121}{3} \approx 0.707\] Now use the inverse cosine to find \(x\): 1. Take the inverse cosine: \[x + 30^\circ = \arccos(0.707)\] 2. Calculate \( \arccos(0.707) \): \[x + 30^\circ \approx 45^\circ\] 3. Solve for \( x \): \[x \approx 45^\circ - 30^\circ \approx 15^\circ\] ### Equation (3) \[2 \tan(2x - 10^\circ) = 3.4641\] Isolate \(\tan\): \[\tan(2x - 10^\circ) = \frac{3.4641}{2} = 1.73205\] Now using the tangent values: 1. Since \(\tan(60^\circ) = \sqrt{3} \approx 1.732\): \[2x - 10^\circ = 60^\circ\] 2. Solve for \(2x\): \[2x = 70^\circ\] 3. Solve for \(x\): \[x = 35^\circ\] ### Equation (4) \[2 \tan(2x) - 10 = 3.4641\] Rearranging gives: \[2 \tan(2x) = 13.4641\] \[\tan(2x) = 6.73205\] Using the tangent values: 1. Since \(\tan(80^\circ) = 5.671\) and close to that with tangent values: \[2x \approx 81^\circ\] 2. Solve for \(2x\): \[2x = 81^\circ\] 3. Solve for \(x\): \[x \approx 40.5^\circ\] ### Summary of Solutions 1. \( x \approx 62.40^\circ \) for equation (1). 2. \( x \approx 15^\circ \) for equation (2). 3. \( x \approx 35^\circ \) for equation (3). 4. \( x \approx 40.5^\circ \) for equation (4).