\( \begin{array}{l} \text { Find the general solutions to these equations. }\\ \begin{array}{ll} 1 & \sin \left(x-16^{\circ}\right)=0,616 \\ 2 & \cos 2 x=0,789 \\ 3 & \cos 3 x=-0,123 \\ 4 & \tan \left(x+56^{\circ}\right)=7,56 \\ 5 & \tan \frac{x}{2}=-1,421 \\ 6 & \sin \left(2 x+44^{\circ}\right)=-0,708 \\ 7 & 3 \cos \left(x-15^{\circ}\right)+1=-0,456 \\ 8 & 2 \tan \left(2 x-10^{\circ}\right)=10,67 \\ \hline \end{array} \end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(\tan\left(x+56\right)=7.56\) - step1: Find the domain: \(\tan\left(x+56\right)=7.56,x\neq \frac{37\pi -112}{2}+k\pi ,k \in \mathbb{Z}\) - step2: Use the inverse trigonometric function: \(x+56=\arctan\left(7.56\right)\) - step3: Add the period: \(x+56=\arctan\left(7.56\right)+k\pi ,k \in \mathbb{Z}\) - step4: Solve the equation: \(x=\arctan\left(7.56\right)-56+k\pi ,k \in \mathbb{Z}\) - step5: Check if the solution is in the defined range: \(x=\arctan\left(7.56\right)-56+k\pi ,k \in \mathbb{Z},x\neq \frac{37\pi -112}{2}+k\pi ,k \in \mathbb{Z}\) - step6: Find the intersection: \(x=\arctan\left(7.56\right)-56+k\pi ,k \in \mathbb{Z}\) - step7: Simplify: \(x\approx -54.560715+k\pi ,k \in \mathbb{Z}\) Solve the equation \( 2\tan(2x-10^{\text{°}})=10.67 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2\tan\left(2x-10\right)=10.67\) - step1: Find the domain: \(2\tan\left(2x-10\right)=10.67,x\neq \frac{\pi +20}{4}+\frac{k\pi }{2},k \in \mathbb{Z}\) - step2: Multiply both sides: \(2\tan\left(2x-10\right)\times \frac{1}{2}=10.67\times \frac{1}{2}\) - step3: Calculate: \(\tan\left(2x-10\right)=10.67\times \frac{1}{2}\) - step4: Calculate: \(\tan\left(2x-10\right)=\frac{1067}{200}\) - step5: Use the inverse trigonometric function: \(2x-10=\arctan\left(\frac{1067}{200}\right)\) - step6: Add the period: \(2x-10=\arctan\left(\frac{1067}{200}\right)+k\pi ,k \in \mathbb{Z}\) - step7: Solve the equation: \(x=\frac{\arctan\left(\frac{1067}{200}\right)+10}{2}+\frac{k\pi }{2},k \in \mathbb{Z}\) - step8: Check if the solution is in the defined range: \(x=\frac{\arctan\left(\frac{1067}{200}\right)+10}{2}+\frac{k\pi }{2},k \in \mathbb{Z},x\neq \frac{\pi +20}{4}+\frac{k\pi }{2},k \in \mathbb{Z}\) - step9: Find the intersection: \(x=\frac{\arctan\left(\frac{1067}{200}\right)+10}{2}+\frac{k\pi }{2},k \in \mathbb{Z}\) Solve the equation \( \cos(2x)=0.789 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(2x\right)=0.789\) - step1: Use the inverse trigonometric function: \(2x=\arccos\left(0.789\right)\) - step2: Calculate: \(\begin{align}&2x=-\arccos\left(0.789\right)\\&2x=\arccos\left(0.789\right)\end{align}\) - step3: Add the period: \(\begin{align}&2x=-\arccos\left(0.789\right)+2k\pi ,k \in \mathbb{Z}\\&2x=\arccos\left(0.789\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step4: Calculate: \(\begin{align}&x=-\frac{\arccos\left(0.789\right)}{2}+k\pi ,k \in \mathbb{Z}\\&2x=\arccos\left(0.789\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Calculate: \(\begin{align}&x=-\frac{\arccos\left(0.789\right)}{2}+k\pi ,k \in \mathbb{Z}\\&x=\frac{\arccos\left(0.789\right)}{2}+k\pi ,k \in \mathbb{Z}\end{align}\) - step6: Find the union: \(x=\left\{ \begin{array}{l}-\frac{\arccos\left(0.789\right)}{2}+k\pi \\\frac{\arccos\left(0.789\right)}{2}+k\pi \end{array}\right.,k \in \mathbb{Z}\) - step7: Simplify: \(x\approx \left\{ \begin{array}{l}-0.330808+k\pi \\0.330808+k\pi \end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( \sin(2x+44^{\text{°}})=-0.708 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sin\left(2x+44\right)=-0.708\) - step1: Use the inverse trigonometric function: \(2x+44=\arcsin\left(-0.708\right)\) - step2: Calculate: \(\begin{align}&2x+44=-\arcsin\left(0.708\right)\\&2x+44=\arcsin\left(0.708\right)+\pi \end{align}\) - step3: Add the period: \(\begin{align}&2x+44=-\arcsin\left(0.708\right)+2k\pi ,k \in \mathbb{Z}\\&2x+44=\arcsin\left(0.708\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step4: Calculate: \(\begin{align}&x=-\frac{\arcsin\left(0.708\right)+44}{2}+k\pi ,k \in \mathbb{Z}\\&2x+44=\arcsin\left(0.708\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Calculate: \(\begin{align}&x=-\frac{\arcsin\left(0.708\right)+44}{2}+k\pi ,k \in \mathbb{Z}\\&x=\frac{\arcsin\left(0.708\right)+\pi -44}{2}+k\pi ,k \in \mathbb{Z}\end{align}\) - step6: Find the union: \(x=\left\{ \begin{array}{l}-\frac{\arcsin\left(0.708\right)+44}{2}+k\pi \\\frac{\arcsin\left(0.708\right)+\pi -44}{2}+k\pi \end{array}\right.,k \in \mathbb{Z}\) - step7: Simplify: \(x\approx \left\{ \begin{array}{l}-22.393331+k\pi \\-20.035873+k\pi \end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( \tan\frac{x}{2}=-1.421 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\tan\left(\frac{x}{2}\right)=-1.421\) - step1: Find the domain: \(\tan\left(\frac{x}{2}\right)=-1.421,x\neq \pi +2k\pi ,k \in \mathbb{Z}\) - step2: Use the inverse trigonometric function: \(\frac{x}{2}=\arctan\left(-1.421\right)\) - step3: Add the period: \(\frac{x}{2}=\arctan\left(-1.421\right)+k\pi ,k \in \mathbb{Z}\) - step4: Solve the equation: \(x=2\arctan\left(-1.421\right)+2k\pi ,k \in \mathbb{Z}\) - step5: Check if the solution is in the defined range: \(x=2\arctan\left(-1.421\right)+2k\pi ,k \in \mathbb{Z},x\neq \pi +2k\pi ,k \in \mathbb{Z}\) - step6: Find the intersection: \(x=2\arctan\left(-1.421\right)+2k\pi ,k \in \mathbb{Z}\) - step7: Simplify: \(x\approx -1.915143+2k\pi ,k \in \mathbb{Z}\) Solve the equation \( \cos(3x)=-0.123 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(3x\right)=-0.123\) - step1: Use the inverse trigonometric function: \(3x=\arccos\left(-0.123\right)\) - step2: Calculate: \(\begin{align}&3x=-\arccos\left(-0.123\right)\\&3x=\arccos\left(-0.123\right)\end{align}\) - step3: Add the period: \(\begin{align}&3x=-\arccos\left(-0.123\right)+2k\pi ,k \in \mathbb{Z}\\&3x=\arccos\left(-0.123\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step4: Calculate: \(\begin{align}&x=-\frac{\arccos\left(-0.123\right)}{3}+\frac{2k\pi }{3},k \in \mathbb{Z}\\&3x=\arccos\left(-0.123\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Calculate: \(\begin{align}&x=-\frac{\arccos\left(-0.123\right)}{3}+\frac{2k\pi }{3},k \in \mathbb{Z}\\&x=\frac{\arccos\left(-0.123\right)}{3}+\frac{2k\pi }{3},k \in \mathbb{Z}\end{align}\) - step6: Find the union: \(x=\left\{ \begin{array}{l}-\frac{\arccos\left(-0.123\right)}{3}+\frac{2k\pi }{3}\\\frac{\arccos\left(-0.123\right)}{3}+\frac{2k\pi }{3}\end{array}\right.,k \in \mathbb{Z}\) - step7: Simplify: \(x\approx \left\{ \begin{array}{l}-0.564703+\frac{2k\pi }{3}\\0.564703+\frac{2k\pi }{3}\end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( 3\cos(x-15^{\text{°}})+1=-0.456 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3\cos\left(x-15\right)+1=-0.456\) - step1: Move the constant to the right side: \(3\cos\left(x-15\right)=-0.456-1\) - step2: Subtract the numbers: \(3\cos\left(x-15\right)=-1.456\) - step3: Divide both sides: \(\frac{3\cos\left(x-15\right)}{3}=\frac{-1.456}{3}\) - step4: Divide the numbers: \(\cos\left(x-15\right)=-\frac{182}{375}\) - step5: Use the inverse trigonometric function: \(x-15=\arccos\left(-\frac{182}{375}\right)\) - step6: Calculate: \(\begin{align}&x-15=-\arccos\left(-\frac{182}{375}\right)\\&x-15=\arccos\left(-\frac{182}{375}\right)\end{align}\) - step7: Add the period: \(\begin{align}&x-15=-\arccos\left(-\frac{182}{375}\right)+2k\pi ,k \in \mathbb{Z}\\&x-15=\arccos\left(-\frac{182}{375}\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step8: Calculate: \(\begin{align}&x=-\arccos\left(-\frac{182}{375}\right)+15+2k\pi ,k \in \mathbb{Z}\\&x-15=\arccos\left(-\frac{182}{375}\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step9: Calculate: \(\begin{align}&x=-\arccos\left(-\frac{182}{375}\right)+15+2k\pi ,k \in \mathbb{Z}\\&x=\arccos\left(-\frac{182}{375}\right)+15+2k\pi ,k \in \mathbb{Z}\end{align}\) - step10: Find the union: \(x=\left\{ \begin{array}{l}-\arccos\left(-\frac{182}{375}\right)+15+2k\pi \\\arccos\left(-\frac{182}{375}\right)+15+2k\pi \end{array}\right.,k \in \mathbb{Z}\) Here are the general solutions for the given equations: 1. **Equation:** \( \sin(x - 16^{\circ}) = 0.616 \) **Solution:** \[ x \approx \left\{ \begin{array}{l} -0.330808 + k\pi \\ 0.330808 + k\pi \end{array} \right., k \in \mathbb{Z} \] 2. **Equation:** \( \cos(2x) = 0.789 \) **Solution:** \[ x \approx \left\{ \begin{array}{l} -0.564703 + \frac{2k\pi}{3} \\ 0.564703 + \frac{2k\pi}{3} \end{array} \right., k \in \mathbb{Z} \] 3. **Equation:** \( \cos(3x) = -0.123 \) **Solution:** \[ x \approx \left\{ \begin{array}{l} -0.564703 + \frac{2k\pi}{3} \\ 0.564703 + \frac{2k\pi}{3} \end{array} \right., k \in \mathbb{Z} \] 4. **Equation:** \( \tan(x + 56^{\circ}) = 7.56 \) **Solution:** \[ x \approx -54.560715 + k\pi, k \in \mathbb{Z} \] 5. **Equation:** \( \tan\left(\frac{x}{2}\right) = -1.421 \) **Solution:** \[ x \approx -1.915143 + 2k\pi, k \in \mathbb{Z} \] 6. **Equation:** \( \sin(2x + 44^{\circ}) = -0.708 \) **Solution:** \[ x \approx \left\{ \begin{array}{l} -22.393331 + k\pi \\ -20.035873 + k\pi \end{array} \right., k \in \mathbb{Z} \] 7. **Equation:** \( 3\cos(x - 15^{\circ}) + 1 = -0.456 \) **Solution:** \[ x = \left\{ \begin{array}{l} -\arccos\left(-\frac{182}{375}\right) + 15 + 2k\pi \\ \arccos\left(-\frac{182}{375}\right) + 15 + 2k\pi \end{array} \right., k \in \mathbb{Z} \] 8. **Equation:** \( 2\tan(2x - 10^{\circ}) = 10.67 \) **Solution:** \[ x = \frac{\arctan\left(\frac{1067}{200}\right) + 10}{2} + \frac{k\pi}{2}, k \in \mathbb{Z} \] These solutions provide the general forms for each equation, where \( k \) is any integer.