etermine the value of the acute angle \( x \) in each of the following equations \( \sin \left(x-20^{\circ}\right)=0,678 \) \( 2 \tan \left(2 x-10^{\circ}\right)=3,4641 \)

To start off with the equation \( \sin \left(x-20^{\circ}\right)=0,678 \), you will first need to find the inverse sine. Thus, we take: \[ x - 20^{\circ} = \sin^{-1}(0.678) \] Calculating the inverse sine gives us approximately \( 42.04^{\circ} \): \[ x - 20^{\circ} \approx 42.04^{\circ} \] Now, solve for \( x \): \[ x \approx 42.04^{\circ} + 20^{\circ} \approx 62.04^{\circ} \] Since sine has another solution for \( \sin \theta = 0.678 \) within the range of \( 0^{\circ} \) to \( 90^{\circ} \), we also take \( 180^{\circ} - 42.04^{\circ} \): \[ x - 20^{\circ} = 180^{\circ} - 42.04^{\circ} \implies x \approx 180^{\circ} - 42.04^{\circ} + 20^{\circ} \approx 157.96^{\circ} \] For the acute angle, we only consider \( 62.04^{\circ} \). Next, for the equation \( 2 \tan \left(2 x-10^{\circ}\right)=3,4641 \), we first divide by 2: \[ \tan \left(2 x - 10^{\circ}\right) = \frac{3,4641}{2} \approx 1.73205 \] Since \( \tan(60^{\circ}) \approx 1.73205 \), we equate: \[ 2 x - 10^{\circ} = 60^{\circ} \implies 2x = 70^{\circ} \implies x = 35^{\circ} \] Thus, the value of the acute angles \( x \) are approximately \( 62.04^{\circ} \) and \( 35^{\circ} \).