Deftermine the size of the acute angle \( \theta \) in each of the following, rounding your answers off to two decimal places where necessary. \( \begin{array}{llll}\text { (1) } \sin \theta=0,866 & \text { (2) } \cos \theta=0,866 & \text { (3) } \tan \theta=1,703\end{array} \) \( \begin{array}{llll}\text { (4) } \sin \theta=1 & \text { (5) } \cos \theta=1 & \text { (6) } \tan \theta=1\end{array} \) Solve the following equations by finding the size of the acute angle \( \theta \) in each case rounding your answers off correct to one decimal place where necessary \( \begin{array}{llll}\text { (1) } \cos 3 \theta=0,33 & \text { (2) } 3 \cos \theta=0,33 & \text { (3) } \sin 4 \theta=0,888\end{array} \) (4) \( 4 \sin \theta=0,888 \) (5) \( \tan 4 \theta=4 \)

Calculate the value by following steps: - step0: Calculate: \(\arccos\left(1\right)\) - step1: Calculate the trigonometric value: \(0\) Solve the equation \( \tan(4\theta)=4 \). Solve the equation by following steps: - step0: Solve for \(\theta\): \(\tan\left(4\theta \right)=4\) - step1: Find the domain: \(\tan\left(4\theta \right)=4,\theta \neq \frac{\pi }{8}+\frac{k\pi }{4},k \in \mathbb{Z}\) - step2: Use the inverse trigonometric function: \(4\theta =\arctan\left(4\right)\) - step3: Add the period: \(4\theta =\arctan\left(4\right)+k\pi ,k \in \mathbb{Z}\) - step4: Solve the equation: \(\theta =\frac{\arctan\left(4\right)}{4}+\frac{k\pi }{4},k \in \mathbb{Z}\) - step5: Check if the solution is in the defined range: \(\theta =\frac{\arctan\left(4\right)}{4}+\frac{k\pi }{4},k \in \mathbb{Z},\theta \neq \frac{\pi }{8}+\frac{k\pi }{4},k \in \mathbb{Z}\) - step6: Find the intersection: \(\theta =\frac{\arctan\left(4\right)}{4}+\frac{k\pi }{4},k \in \mathbb{Z}\) Calculate or simplify the expression \( \arctan(1) \). Calculate the value by following steps: - step0: Calculate: \(\arctan\left(1\right)\) - step1: Calculate the trigonometric value: \(\frac{\pi }{4}\) Solve the equation \( 3\cos(\theta)=0.33 \). Solve the equation by following steps: - step0: Solve for \(\theta\): \(3\cos\left(\theta \right)=0.33\) - step1: Multiply both sides: \(3\cos\left(\theta \right)\times \frac{1}{3}=0.33\times \frac{1}{3}\) - step2: Calculate: \(\cos\left(\theta \right)=0.33\times \frac{1}{3}\) - step3: Calculate: \(\cos\left(\theta \right)=\frac{11}{100}\) - step4: Use the inverse trigonometric function: \(\theta =\arccos\left(\frac{11}{100}\right)\) - step5: Calculate: \(\begin{align}&\theta =-\arccos\left(\frac{11}{100}\right)\\&\theta =\arccos\left(\frac{11}{100}\right)\end{align}\) - step6: Add the period: \(\begin{align}&\theta =-\arccos\left(\frac{11}{100}\right)+2k\pi ,k \in \mathbb{Z}\\&\theta =\arccos\left(\frac{11}{100}\right)+2k\pi ,k \in \mathbb{Z}\end{align}\) - step7: Find the union: \(\theta =\left\{ \begin{array}{l}-\arccos\left(\frac{11}{100}\right)+2k\pi \\\arccos\left(\frac{11}{100}\right)+2k\pi \end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( 4\sin(\theta)=0.888 \). Solve the equation by following steps: - step0: Solve for \(\theta\): \(4\sin\left(\theta \right)=0.888\) - step1: Multiply both sides: \(4\sin\left(\theta \right)\times \frac{1}{4}=0.888\times \frac{1}{4}\) - step2: Calculate: \(\sin\left(\theta \right)=0.888\times \frac{1}{4}\) - step3: Calculate: \(\sin\left(\theta \right)=\frac{111}{500}\) - step4: Use the inverse trigonometric function: \(\theta =\arcsin\left(\frac{111}{500}\right)\) - step5: Calculate: \(\begin{align}&\theta =\arcsin\left(\frac{111}{500}\right)\\&\theta =-\arcsin\left(\frac{111}{500}\right)+\pi \end{align}\) - step6: Add the period: \(\begin{align}&\theta =\arcsin\left(\frac{111}{500}\right)+2k\pi ,k \in \mathbb{Z}\\&\theta =-\arcsin\left(\frac{111}{500}\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step7: Find the union: \(\theta =\left\{ \begin{array}{l}\arcsin\left(\frac{111}{500}\right)+2k\pi \\-\arcsin\left(\frac{111}{500}\right)+\pi +2k\pi \end{array}\right.,k \in \mathbb{Z}\) Calculate or simplify the expression \( \arccos(0.866) \). Calculate the value by following steps: - step0: Calculate: \(\arccos\left(0.866\right)\) Solve the equation \( \sin(4\theta)=0.888 \). Solve the equation by following steps: - step0: Solve for \(\theta\): \(\sin\left(4\theta \right)=0.888\) - step1: Use the inverse trigonometric function: \(4\theta =\arcsin\left(0.888\right)\) - step2: Calculate: \(\begin{align}&4\theta =\arcsin\left(0.888\right)\\&4\theta =-\arcsin\left(0.888\right)+\pi \end{align}\) - step3: Add the period: \(\begin{align}&4\theta =\arcsin\left(0.888\right)+2k\pi ,k \in \mathbb{Z}\\&4\theta =-\arcsin\left(0.888\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step4: Calculate: \(\begin{align}&\theta =\frac{\arcsin\left(0.888\right)}{4}+\frac{k\pi }{2},k \in \mathbb{Z}\\&4\theta =-\arcsin\left(0.888\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Calculate: \(\begin{align}&\theta =\frac{\arcsin\left(0.888\right)}{4}+\frac{k\pi }{2},k \in \mathbb{Z}\\&\theta =\frac{-\arcsin\left(0.888\right)+\pi }{4}+\frac{k\pi }{2},k \in \mathbb{Z}\end{align}\) - step6: Find the union: \(\theta =\left\{ \begin{array}{l}\frac{\arcsin\left(0.888\right)}{4}+\frac{k\pi }{2}\\\frac{-\arcsin\left(0.888\right)+\pi }{4}+\frac{k\pi }{2}\end{array}\right.,k \in \mathbb{Z}\) - step7: Simplify: \(\theta \approx \left\{ \begin{array}{l}0.273244+\frac{k\pi }{2}\\0.512154+\frac{k\pi }{2}\end{array}\right.,k \in \mathbb{Z}\) Calculate or simplify the expression \( \arcsin(0.866) \). Calculate the value by following steps: - step0: Calculate: \(\arcsin\left(0.866\right)\) Calculate or simplify the expression \( \arctan(1.703) \). Calculate the value by following steps: - step0: Calculate: \(\arctan\left(1.703\right)\) Calculate or simplify the expression \( \arcsin(1) \). Calculate the value by following steps: - step0: Calculate: \(\arcsin\left(1\right)\) - step1: Calculate the trigonometric value: \(\frac{\pi }{2}\) Let's solve the problems step by step. ### Finding the Acute Angles 1. **For \( \sin \theta = 0.866 \)**: \[ \theta = \arcsin(0.866) \approx 1.05 \text{ radians} \approx 60^\circ \] 2. **For \( \cos \theta = 0.866 \)**: \[ \theta = \arccos(0.866) \approx 0.52 \text{ radians} \approx 30^\circ \] 3. **For \( \tan \theta = 1.703 \)**: \[ \theta = \arctan(1.703) \approx 1.04 \text{ radians} \approx 59.9^\circ \] 4. **For \( \sin \theta = 1 \)**: \[ \theta = \arcsin(1) = \frac{\pi}{2} \approx 1.57 \text{ radians} \approx 90^\circ \] 5. **For \( \cos \theta = 1 \)**: \[ \theta = \arccos(1) = 0 \text{ radians} \approx 0^\circ \] 6. **For \( \tan \theta = 1 \)**: \[ \theta = \arctan(1) = \frac{\pi}{4} \approx 0.79 \text{ radians} \approx 45^\circ \] ### Solving the Equations 1. **For \( \cos 3\theta = 0.33 \)**: \[ \theta = \left\{ -\arccos\left(\frac{11}{100}\right) + 2k\pi, \arccos\left(\frac{11}{100}\right) + 2k\pi \right\}, k \in \mathbb{Z} \] Approximating gives: \[ \theta \approx \left\{ -1.46 + 2k\pi, 1.46 + 2k\pi \right\} \] 2. **For \( 3 \cos \theta = 0.33 \)**: \[ \theta \approx \left\{ 0.27 + \frac{k\pi}{2}, 0.51 + \frac{k\pi}{2} \right\}, k \in \mathbb{Z} \] 3. **For \( \sin 4\theta = 0.888 \)**: \[ \theta = \left\{ \arcsin\left(\frac{111}{500}\right) + 2k\pi, -\arcsin\left(\frac{111}{500}\right) + \pi + 2k\pi \right\}, k \in \mathbb{Z} \] Approximating gives: \[ \theta \approx \left\{ 0.22 + 2k\pi, 2.92 + 2k\pi \right\} \] 4. **For \( 4 \sin \theta = 0.888 \)**: \[ \theta = \frac{1}{4} \arcsin\left(0.888\right) + \frac{k\pi}{4}, k \in \mathbb{Z} \] Approximating gives: \[ \theta \approx 0.33 + \frac{k\pi}{4}, k \in \mathbb{Z} \] 5. **For \( \tan 4\theta = 4 \)**: \[ \theta = \frac{1}{4} \arctan(4) + \frac{k\pi}{4}, k \in \mathbb{Z} \] Approximating gives: \[ \theta \approx 0.33 + \frac{k\pi}{4}, k \in \mathbb{Z} \] ### Summary of Results - **Acute Angles**: 1. \( \theta \approx 60^\circ \) 2. \( \theta \approx 30^\circ \) 3. \( \theta \approx 59.9^\circ \) 4. \( \theta \approx 90^\circ \) 5. \( \theta \approx 0^\circ \) 6. \( \theta \approx 45^\circ \) - **Equations**: 1. \( \theta \approx \left\{ -1.46 + 2k\pi, 1.46 + 2k\pi \right\} \) 2. \( \theta \approx \left\{ 0.27 + \frac{k\pi}{2}, 0.51 + \frac{k\pi}{2} \right\} \) 3. \( \theta \approx \left\{ 0.22 + 2k\pi, 2.92 + 2k\pi \right\} \) 4. \( \theta \approx 0.33 + \frac{k\pi}{4} \) 5. \( \theta \approx 0.33 + \frac{k\pi}{4} \)