Qurition 4 ivlve for \( x \) where \( 0^{\circ} \leq x \leq 90^{\circ} \), in each of the following equations. Give your answer correct. 1) TWO decimal places. \[ \begin{array}{l} \text { 4.shl } \cos 3 x=5,75 \\ 4 . t^{2} \cdot \operatorname{cosec}\left(x-355^{\circ}\right)=7,814 \quad \text { Classwoik } \\ \text { 4ol3, } \frac{\cos \left(2 x+10^{\circ}\right)}{3}=0,159 \end{array} \] 1 \( A=70^{\circ} \) and \( B=40^{\circ} \), use your calculator to evaluate the following(correct to ONE decimal place). \( \therefore 1 \quad \tan (A+B) \) .2. \( 2 \sin ^{2} A \) \( \therefore 3 \sqrt[3]{2 \sec \frac{3 B}{2}} \) thour the use of the cylculator, showing all your working, determine the value of: \( \frac{.11^{2} 60 \cdot \tan 45^{\circ} \cdot \sin 30^{\circ}}{\cos 30^{\circ}} \)

Alright, I need to solve the given trigonometric equations and perform some calculations using a calculator. Let's break down the problem step by step. **First Equation:** \[ 4 \cdot \sin 3x = 5.75 \] To solve for \( x \), I'll start by isolating \( \sin 3x \): \[ \sin 3x = \frac{5.75}{4} = 1.4375 \] Wait a minute, the sine of an angle cannot exceed 1. This suggests that there might be a mistake in the equation or perhaps a typo. Maybe it's supposed to be \( 4 \cdot \sin 3x = 5.75 \), but even then, \( \sin 3x = 1.4375 \) is impossible. I'll proceed under the assumption that the equation is correct and see where it leads. Using the inverse sine function: \[ 3x = \arcsin(1.4375) \] But since \( \arcsin \) is only defined for values between -1 and 1, this equation has no real solution. Perhaps there's an error in the problem statement. **Second Equation:** \[ 4 \cdot t^{2} \cdot \operatorname{cosec}\left(x - 355^{\circ}\right) = 7.814 \] First, let's simplify the cosecant function: \[ \operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)} \] So, \[ 4 \cdot t^{2} \cdot \frac{1}{\sin\left(x - 355^{\circ}\right)} = 7.814 \] Assuming \( t \) is a constant or another variable, I'll focus on solving for \( x \). Rearranging the equation: \[ \sin\left(x - 355^{\circ}\right) = \frac{4 \cdot t^{2}}{7.814} \] Again, without knowing the value of \( t \), I can't proceed further. This equation also seems incomplete or missing information. **Third Equation:** \[ 4 \cdot \frac{\cos\left(2x + 10^{\circ}\right)}{3} = 0.159 \] Solving for \( \cos\left(2x + 10^{\circ}\right) \): \[ \cos\left(2x + 10^{\circ}\right) = \frac{0.159 \cdot 3}{4} = 0.11925 \] Now, using the inverse cosine function: \[ 2x + 10^{\circ} = \arccos(0.11925) \] Calculating \( \arccos(0.11925) \): \[ \arccos(0.11925) \approx 83.62^{\circ} \] So, \[ 2x = 83.62^{\circ} - 10^{\circ} = 73.62^{\circ} \] \[ x = \frac{73.62^{\circ}}{2} \approx 36.81^{\circ} \] Rounding to two decimal places: \[ x \approx 36.81^{\circ} \] **Calculator Evaluations:** 1. **Evaluate \( \tan(A + B) \) where \( A = 70^{\circ} \) and \( B = 40^{\circ} \):** \[ A + B = 70^{\circ} + 40^{\circ} = 110^{\circ} \] \[ \tan(110^{\circ}) \approx -2.747 \] Rounded to one decimal place: \[ \tan(110^{\circ}) \approx -2.7 \] 2. **Evaluate \( 2 \sin^{2} A \):** \[ \sin(70^{\circ}) \approx 0.9397 \] \[ \sin^{2}(70^{\circ}) \approx (0.9397)^2 \approx 0.882 \] \[ 2 \cdot 0.882 \approx 1.764 \] Rounded to one decimal place: \[ 2 \sin^{2}(70^{\circ}) \approx 1.8 \] 3. **Evaluate \( \sqrt[3]{2 \sec \frac{3B}{2}} \):** \[ \frac{3B}{2} = \frac{3 \cdot 40^{\circ}}{2} = 60^{\circ} \] \[ \sec(60^{\circ}) = \frac{1}{\cos(60^{\circ})} = \frac{1}{0.5} = 2 \] \[ 2 \sec(60^{\circ}) = 2 \cdot 2 = 4 \] \[ \sqrt[3]{4} \approx 1.587 \] Rounded to one decimal place: \[ \sqrt[3]{2 \sec \frac{3B}{2}} \approx 1.6 \] **Final Calculation:** \[ \frac{0.11^{2} \cdot 60 \cdot \tan 45^{\circ} \cdot \sin 30^{\circ}}{\cos 30^{\circ}} \] First, calculate each component: \[ 0.11^{2} = 0.0121 \] \[ \tan 45^{\circ} = 1 \] \[ \sin 30^{\circ} = 0.5 \] \[ \cos 30^{\circ} = 0.866 \] Now, plug these values into the equation: \[ \frac{0.0121 \cdot 60 \cdot 1 \cdot 0.5}{0.866} = \frac{0.0121 \cdot 30}{0.866} = \frac{0.363}{0.866} \approx 0.42 \] Rounded to two decimal places: \[ \approx 0.42 \]