quadrant, \( R \hat{O Q}=Q \) and \( 13 \sin Q=1 \alpha \). Make use of the diagram abone to: \( 5.1, y) \) determine the value of \( x \). 52 . calculate the value of \( \sec Q \) 53 calculate the value of \( 5 \tan \theta+1 \)

To solve the problem, let's break it down step by step based on the information provided. ### Known Conditions 1. We have a right triangle in quadrant \( R \) with angle \( Q \). 2. The equation \( 13 \sin Q = 1 \alpha \) suggests a relationship involving the sine of angle \( Q \). 3. We need to determine the value of \( x \) and calculate \( \sec Q \) and \( 5 \tan \theta + 1 \). ### Step 1: Determine \( \sin Q \) From the equation \( 13 \sin Q = 1 \alpha \), we can express \( \sin Q \) as: \[ \sin Q = \frac{1 \alpha}{13} \] ### Step 2: Calculate \( \sec Q \) The secant function is defined as: \[ \sec Q = \frac{1}{\cos Q} \] Using the Pythagorean identity: \[ \sin^2 Q + \cos^2 Q = 1 \] we can find \( \cos Q \) once we have \( \sin Q \). ### Step 3: Calculate \( \tan Q \) The tangent function is defined as: \[ \tan Q = \frac{\sin Q}{\cos Q} \] ### Step 4: Calculate \( 5 \tan \theta + 1 \) We will need to know the value of \( \tan Q \) to compute this expression. ### Calculations Let's perform the calculations for \( \sec Q \) and \( 5 \tan Q + 1 \) using the known relationships. First, we need to find \( \cos Q \) using the sine value. 1. **Calculate \( \cos Q \)**: \[ \cos^2 Q = 1 - \sin^2 Q \] 2. **Calculate \( \sec Q \)**: \[ \sec Q = \frac{1}{\cos Q} \] 3. **Calculate \( \tan Q \)**: \[ \tan Q = \frac{\sin Q}{\cos Q} \] 4. **Calculate \( 5 \tan Q + 1 \)**: \[ 5 \tan Q + 1 \] Let's proceed with the calculations. Since we don't have a specific value for \( \alpha \), we will express everything in terms of \( \alpha \). ### Tool Calls I will calculate \( \sec Q \) and \( 5 \tan Q + 1 \) based on the expressions derived. Let's assume \( \alpha = 1 \) for simplicity in calculations. 1. Calculate \( \sin Q \): \[ \sin Q = \frac{1}{13} \] 2. Calculate \( \cos Q \): \[ \cos^2 Q = 1 - \left(\frac{1}{13}\right)^2 \] 3. Calculate \( \sec Q \): \[ \sec Q = \frac{1}{\cos Q} \] 4. Calculate \( \tan Q \): \[ \tan Q = \frac{\sin Q}{\cos Q} \] 5. Calculate \( 5 \tan Q + 1 \): \[ 5 \tan Q + 1 \] Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{13}\) Calculate or simplify the expression \( 1 - (1/13)^2 \). Calculate the value by following steps: - step0: Calculate: \(1-\left(\frac{1}{13}\right)^{2}\) - step1: Rewrite the expression: \(1-\frac{1}{13^{2}}\) - step2: Evaluate the power: \(1-\frac{1}{169}\) - step3: Reduce fractions to a common denominator: \(\frac{169}{169}-\frac{1}{169}\) - step4: Transform the expression: \(\frac{169-1}{169}\) - step5: Subtract the numbers: \(\frac{168}{169}\) Calculate or simplify the expression \( \sqrt(1 - (1/13)^2) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{1-\left(\frac{1}{13}\right)^{2}}\) - step1: Subtract the numbers: \(\sqrt{\frac{168}{169}}\) - step2: Use the properties of radicals: \(\frac{\sqrt{168}}{\sqrt{169}}\) - step3: Simplify the expression: \(\frac{2\sqrt{42}}{13}\) Calculate or simplify the expression \( (1/13)/(\sqrt(1 - (1/13)^2)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{1}{13}}{\left(\sqrt{1-\left(\frac{1}{13}\right)^{2}}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{1}{13}}{\sqrt{1-\left(\frac{1}{13}\right)^{2}}}\) - step2: Subtract the numbers: \(\frac{\frac{1}{13}}{\sqrt{\frac{168}{169}}}\) - step3: Simplify the root: \(\frac{\frac{1}{13}}{\frac{2\sqrt{42}}{13}}\) - step4: Multiply by the reciprocal: \(\frac{1}{13}\times \frac{13}{2\sqrt{42}}\) - step5: Reduce the numbers: \(1\times \frac{1}{2\sqrt{42}}\) - step6: Multiply the fractions: \(\frac{1}{2\sqrt{42}}\) - step7: Multiply by the Conjugate: \(\frac{\sqrt{42}}{2\sqrt{42}\times \sqrt{42}}\) - step8: Multiply the numbers: \(\frac{\sqrt{42}}{84}\) Calculate or simplify the expression \( 1/(\sqrt(1 - (1/13)^2)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{\left(\sqrt{1-\left(\frac{1}{13}\right)^{2}}\right)}\) - step1: Remove the parentheses: \(\frac{1}{\sqrt{1-\left(\frac{1}{13}\right)^{2}}}\) - step2: Subtract the numbers: \(\frac{1}{\sqrt{\frac{168}{169}}}\) - step3: Simplify the root: \(\frac{1}{\frac{2\sqrt{42}}{13}}\) - step4: Multiply by the reciprocal: \(\frac{13}{2\sqrt{42}}\) - step5: Multiply by the Conjugate: \(\frac{13\sqrt{42}}{2\sqrt{42}\times \sqrt{42}}\) - step6: Multiply the numbers: \(\frac{13\sqrt{42}}{84}\) Calculate or simplify the expression \( 5*((1/13)/(\sqrt(1 - (1/13)^2))) + 1 \). Calculate the value by following steps: - step0: Calculate: \(5\left(\frac{\frac{1}{13}}{\left(\sqrt{1-\left(\frac{1}{13}\right)^{2}}\right)}\right)+1\) - step1: Remove the parentheses: \(5\left(\frac{\frac{1}{13}}{\sqrt{1-\left(\frac{1}{13}\right)^{2}}}\right)+1\) - step2: Subtract the numbers: \(5\left(\frac{\frac{1}{13}}{\sqrt{\frac{168}{169}}}\right)+1\) - step3: Simplify the root: \(5\left(\frac{\frac{1}{13}}{\frac{2\sqrt{42}}{13}}\right)+1\) - step4: Divide the terms: \(5\times \frac{1}{2\sqrt{42}}+1\) - step5: Multiply the fractions: \(\frac{5}{2\sqrt{42}}+1\) - step6: Calculate: \(\frac{5\sqrt{42}}{84}+1\) - step7: Reduce fractions to a common denominator: \(\frac{5\sqrt{42}}{84}+\frac{84}{84}\) - step8: Transform the expression: \(\frac{5\sqrt{42}+84}{84}\) ### Summary of Calculations 1. **Calculate \( \sin Q \)**: \[ \sin Q = \frac{1}{13} \approx 0.076923 \] 2. **Calculate \( \cos Q \)**: \[ \cos^2 Q = 1 - \left(\frac{1}{13}\right)^2 = \frac{168}{169} \quad \Rightarrow \quad \cos Q = \sqrt{\frac{168}{169}} \approx 0.994083 \] 3. **Calculate \( \sec Q \)**: \[ \sec Q = \frac{1}{\cos Q} = \frac{13\sqrt{42}}{84} \approx 1.002972 \] 4. **Calculate \( \tan Q \)**: \[ \tan Q = \frac{\sin Q}{\cos Q} = \frac{\frac{1}{13}}{\sqrt{1 - \left(\frac{1}{13}\right)^2}} = \frac{\sqrt{42}}{84} \approx 0.077152 \] 5. **Calculate \( 5 \tan Q + 1 \)**: \[ 5 \tan Q + 1 = 5 \cdot \frac{\sqrt{42}}{84} + 1 = \frac{5\sqrt{42} + 84}{84} \approx 1.385758 \] ### Final Results - The value of \( x \) is not explicitly defined in the problem, but we have calculated the necessary trigonometric values. - \( \sec Q \approx 1.002972 \) - \( 5 \tan Q + 1 \approx 1.385758 \) If you have any further questions or need additional calculations, feel free to ask!