NursTION 4 Solve for \( x \) where \( 0^{\prime} \leq x \leq 90^{\circ} \), in each of the following equations. Give your answer correct. *) TWO decimal places. \( 1.11 \tan 3 x=5,75 \) .1.2 \( \operatorname{cosec}\left(x-35^{\circ}\right)=7,814 \) \( .13 \frac{\cos \left(2 x+10^{\circ}\right)}{3}=0,159 \) 1 \( A=70^{\circ} \) and \( B=40^{\circ} \), use your calculator to evaluate the following(correct to ONE decimal piace). \( 2.1 \quad \tan (A+B) \) \( 2.2 \sin ^{2} A \) \( \therefore 3 \sqrt{2 \sec \frac{3 B}{2}} \) thivut the use of the calculator, showing all your working, determine the value of: \[ \frac{11^{2} 60 \cdot \tan 45^{\circ} \cdot \sin 30^{\circ}}{\cos n 0^{\circ}} \] TOTA
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To solve the equations: 1. **For** \( 1.11 \tan 3x = 5.75 \): Rearranging gives \( \tan 3x = \frac{5.75}{1.11} \approx 5.18 \). Now, find \( 3x \) by taking the arctan, \( 3x = \tan^{-1}(5.18) \). This gives \( 3x \approx 78.58^\circ \), so \( x \approx 26.19^\circ \) (two decimal places). 2. **For** \( \operatorname{cosec}(x - 35^\circ) = 7.814 \): This implies \( \sin(x - 35^\circ) = \frac{1}{7.814} \approx 0.128 \). Thus, \( x - 35^\circ \approx \sin^{-1}(0.128) \approx 7.37^\circ \). Therefore, \( x \approx 42.37^\circ \) (two decimal places). 3. **For** \( \frac{\cos(2x + 10^\circ)}{3} = 0.159 \): Rearranging yields \( \cos(2x + 10^\circ) = 0.477 \). This leads to \( 2x + 10^\circ \approx 61.80^\circ \) or \( 298.20^\circ \). Thus, \( x \approx 25.90^\circ \) or \( 144.10^\circ \) (the latter is out of bounds). Keeping the valid solution: \( x \approx 25.90^\circ \) (two decimal places). Now, using \( A = 70^\circ \) and \( B = 40^\circ \): 1. **For** \( \tan(A+B) = \tan(110^\circ) \approx -0.364 \) (correct to one decimal place). 2. **For** \( \sin^2 A = \sin^2(70^\circ) \approx 0.75 \) (correct to one decimal place). 3. **For** \( 3\sqrt{2 \sec \frac{3B}{2}} = 3\sqrt{2 \sec(60^\circ)} \approx 7.35 \) (correct to one decimal place). Finally, to compute \( \frac{11^{2} \cdot 60 \cdot \tan 45^{\circ} \cdot \sin 30^{\circ}}{\cos 0^\circ} \): - \( \tan 45^\circ = 1 \), \( \sin 30^\circ = 0.5 \), \( \cos 0^\circ = 1 \). - Therefore, the expression simplifies to \( \frac{121 \cdot 60 \cdot 1 \cdot 0.5}{1} = 3630 \). Thus, your answers are \( x \approx 26.19^\circ, 42.37^\circ, 25.90^\circ \) respectively, while for the additional calculations, you have evaluations for each mathematical operation listed.
