If \( \sin x=\frac{7}{25} \) and \( 180^{\circ}

Ask by Hilton King. in South Africa

Mar 14,2025

To solve the problem, we will break it down into two parts. ### Part 1: Finding \( 25 \cos x - \frac{7}{\tan x} \) 1. **Given**: \( \sin x = \frac{7}{25} \) and \( 180^{\circ} < x < 360^{\circ} \). 2. **Find \( \cos x \)**: - Since \( x \) is in the fourth quadrant, \( \cos x \) is positive. - We can use the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] - Substituting \( \sin x \): \[ \left(\frac{7}{25}\right)^2 + \cos^2 x = 1 \] \[ \frac{49}{625} + \cos^2 x = 1 \] \[ \cos^2 x = 1 - \frac{49}{625} = \frac{625 - 49}{625} = \frac{576}{625} \] - Therefore, \( \cos x = \sqrt{\frac{576}{625}} = \frac{24}{25} \). 3. **Find \( \tan x \)**: - Using the definition of tangent: \[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{24} \] 4. **Calculate \( 25 \cos x - \frac{7}{\tan x} \)**: - Substitute \( \cos x \) and \( \tan x \): \[ 25 \cos x = 25 \cdot \frac{24}{25} = 24 \] \[ \frac{7}{\tan x} = \frac{7}{\frac{7}{24}} = 24 \] - Therefore: \[ 25 \cos x - \frac{7}{\tan x} = 24 - 24 = 0 \] ### Part 2: Proving \( 744 \sin^2 y + \frac{7}{\tan^2 x} = 34 \) 1. **Given**: \( \cos x = \frac{3}{4} \) and \( \tan y = \frac{5}{7} \). 2. **Find \( \tan^2 x \)**: - Using the identity \( \tan x = \frac{\sin x}{\cos x} \): \[ \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \] - We already have \( \sin^2 x = \left(\frac{7}{25}\right)^2 = \frac{49}{625} \) and \( \cos^2 x = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \). - Therefore: \[ \tan^2 x = \frac{\frac{49}{625}}{\frac{9}{16}} = \frac{49 \cdot 16}{625 \cdot 9} = \frac{784}{5625} \] 3. **Calculate \( \frac{7}{\tan^2 x} \)**: \[ \frac{7}{\tan^2 x} = \frac{7 \cdot 5625}{784} = \frac{39375}{784} \] 4. **Find \( \sin^2 y \)**: - Since \( \tan y = \frac{5}{7} \): \[ \sin^2 y = \frac{\tan^2 y}{1 + \tan^2 y} = \frac{\left(\frac{5}{7}\right)^2}{1 + \left(\frac{5}{7}\right)^2} = \frac{\frac{25}{49}}{1 + \frac{25}{49}} = \frac{\frac{25}{49}}{\frac{74}{49}} = \frac{25}{74} \] 5. **Calculate \( 744 \sin^2 y \)**: \[ 744 \sin^2 y = 744 \cdot \frac{25}{74} = 250 \] 6. **Combine the results**: \[ 744 \sin^2 y + \frac{7}{\tan^2 x} = 250 + \frac{39375}{784} \] - To combine, we need a common denominator: \[ 250 = \frac{250 \cdot 784}{784} = \frac{196000}{784} \] - Therefore: \[ 744 \sin^2 y + \frac{7}{\tan^2 x} = \frac{196000 + 39375}{784} = \frac{235375}{784} \] 7. **Check if it equals 34**: - We need to check if \( \frac{235375}{784} = 34 \): \[ 34 \cdot 784 = 26656 \] - Since \( 235375 \neq 26656 \), the equation does not hold. ### Conclusion - The value of \( 25 \cos x - \frac{7}{\tan x} \) is \( 0 \). - The proof for \( 744 \sin^2 y + \frac{7}{\tan^2 x} = 34 \) does not hold as shown in the calculations.