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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

Ask by Young Vaughan. in South Africa
Mar 09,2025

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Solutions for \( x \): - \( x \approx 26.2^{\circ} \) - \( x \approx 42.3^{\circ} \) - \( x \approx 26.1^{\circ} \) Evaluated values: - \( \tan(A + B) \approx -2.747 \) - \( \sin^2(A) \approx 0.883 \) The expression \( \frac{11^2 \cdot 60 \cdot \tan(45^{\circ}) \cdot \sin(30^{\circ})}{\cos(90^{\circ})} \) is undefined due to division by zero.

Solución

Let's solve the equations step by step. ### Part 1: Solve for \( x \) #### 1.1 \( 1.11 \tan(3x) = 5.75 \) 1. **Isolate \( \tan(3x) \)**: \[ \tan(3x) = \frac{5.75}{1.11} \] 2. **Calculate \( \tan(3x) \)**: \[ \tan(3x) \approx 5.17567567568 \] 3. **Find \( 3x \)**: \[ 3x = \tan^{-1}(5.17567567568) \] 4. **Calculate \( 3x \)**: \[ 3x \approx 78.6^{\circ} \] 5. **Solve for \( x \)**: \[ x \approx \frac{78.6^{\circ}}{3} \approx 26.2^{\circ} \] #### 1.2 \( \operatorname{cosec}(x - 35^{\circ}) = 7.814 \) 1. **Convert cosecant to sine**: \[ \sin(x - 35^{\circ}) = \frac{1}{7.814} \] 2. **Calculate \( \sin(x - 35^{\circ}) \)**: \[ \sin(x - 35^{\circ}) \approx 0.127 \] 3. **Find \( x - 35^{\circ} \)**: \[ x - 35^{\circ} = \sin^{-1}(0.127) \] 4. **Calculate \( x - 35^{\circ} \)**: \[ x - 35^{\circ} \approx 7.3^{\circ} \] 5. **Solve for \( x \)**: \[ x \approx 7.3^{\circ} + 35^{\circ} \approx 42.3^{\circ} \] #### 1.3 \( \frac{\cos(2x + 10^{\circ})}{3} = 0.159 \) 1. **Isolate \( \cos(2x + 10^{\circ}) \)**: \[ \cos(2x + 10^{\circ}) = 0.159 \times 3 \] 2. **Calculate \( \cos(2x + 10^{\circ}) \)**: \[ \cos(2x + 10^{\circ}) \approx 0.477 \] 3. **Find \( 2x + 10^{\circ} \)**: \[ 2x + 10^{\circ} = \cos^{-1}(0.477) \] 4. **Calculate \( 2x + 10^{\circ} \)**: \[ 2x + 10^{\circ} \approx 62.2^{\circ} \] 5. **Solve for \( x \)**: \[ 2x \approx 62.2^{\circ} - 10^{\circ} \approx 52.2^{\circ} \] \[ x \approx \frac{52.2^{\circ}}{2} \approx 26.1^{\circ} \] ### Summary of Solutions for \( x \): - \( x \approx 26.2^{\circ} \) (from 1.1) - \( x \approx 42.3^{\circ} \) (from 1.2) - \( x \approx 26.1^{\circ} \) (from 1.3) ### Part 2: Evaluate Trigonometric Functions #### 2.1 \( \tan(A + B) \) 1. **Calculate \( A + B \)**: \[ A + B = 70^{\circ} + 40^{\circ} = 110^{\circ} \] 2. **Find \( \tan(110^{\circ}) \)**: \[ \tan(110^{\circ}) \approx -2.747 \] #### 2.2 \( \sin^2(A) \) 1. **Calculate \( \sin(70^{\circ}) \)**: \[ \sin(70^{\circ}) \approx 0.9397 \] 2. **Find \( \sin^2(70^{\circ}) \)**: \[ \sin^2(70^{\circ}) \approx (0.9397)^2 \approx 0.883 \] #### 2.3 \( 3\sqrt{2 \sec\left(\frac{3B}{2}\right)} \) 1. **Calculate \( \frac{3B}{2} \)**: \[ \frac{3B}{2} = \frac{3 \times 40^{\circ}}{2} = 60^{\circ} \] 2. **Find \( \sec(60^{\circ}) \)**: \[ \sec(60^{\circ}) = \frac{1}{\cos(60^{\circ})} = 2 \] 3. **Calculate \( 3\sqrt{2 \cdot 2} \)**: \[ 3\sqrt{4} = 3 \times 2 = 6 \] ### Part 3: Calculate the Expression \[ \frac{11^2 \cdot 60 \cdot \tan(45^{\circ}) \cdot \sin(30^{\circ})}{\cos(90^{\circ})} \] 1. **Calculate each component**: - \( 11^2 = 121 \) - \( \tan(45^{\circ}) = 1 \) - \( \sin(30^{\circ}) = 0.5 \) - \( \cos(90^{\circ}) = 0 \) 2. **Substitute into the expression**: \[ \frac{121 \cdot 60 \cdot 1 \cdot 0.5}{0} \] Since division by zero is undefined, the expression does not yield a valid numerical result. ### Final Summary: - Solutions for \( x \): - \( x \approx 26.2^{\circ} \) - \( x \approx 42.3^{\circ} \) - \( x \approx 26.1^{\circ} \) - Evaluated values: - \( \tan(A + B) \approx -2.747 \) - \( \sin^2(A) \approx 0.883

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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.

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