Puation 5 5.1 If \( 5 \sin x=4 \) and \( x \in\left[0^{\circ} ; 90^{\circ}\right] \); use the sketch to calculate the value the following: \[ 5.12 \sin ^{2} x+\cos ^{2} x \] 5.2 Solve for \( x \) and \( x \in\left[0^{\circ} ; 90^{\circ}\right] \) : \[ \sin x=0,3 \]

Let's solve the problem step by step. ### 5.1 Calculate \( 5 \sin^2 x + \cos^2 x \) We start with the equation given: \[ 5 \sin x = 4 \] From this, we can find \( \sin x \): \[ \sin x = \frac{4}{5} \] Next, we can find \( \cos^2 x \) using the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] First, we calculate \( \sin^2 x \): \[ \sin^2 x = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \] Now, we can find \( \cos^2 x \): \[ \cos^2 x = 1 - \sin^2 x = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \] Now we can substitute \( \sin^2 x \) and \( \cos^2 x \) into the expression \( 5 \sin^2 x + \cos^2 x \): \[ 5 \sin^2 x + \cos^2 x = 5 \left(\frac{16}{25}\right) + \frac{9}{25} \] Calculating this gives: \[ = \frac{80}{25} + \frac{9}{25} = \frac{89}{25} \] ### 5.2 Solve for \( x \) where \( \sin x = 0.3 \) To find \( x \) in the interval \( [0^{\circ}, 90^{\circ}] \), we can use the inverse sine function: \[ x = \sin^{-1}(0.3) \] Now, let's calculate \( x \) using the inverse sine function. I'll perform the calculation for \( x \). Calculate the value by following steps: - step0: Calculate: \(\arcsin\left(0.3\right)\) The value of \( x \) can be calculated as: \[ x = \arcsin(0.3) \] Now, let's compute the numerical value of \( \arcsin(0.3) \) in degrees. Calculating this gives: \[ x \approx 17.46^{\circ} \] ### Summary of Results - For \( 5.1 \): \[ 5 \sin^2 x + \cos^2 x = \frac{89}{25} \] - For \( 5.2 \): \[ x \approx 17.46^{\circ} \]