Pregunta
upstudy study bank question image url

If \( \tan \theta=\frac{12}{5} \) and \( 90^{\circ}
Ask by Franklin Gibbs. in South Africa
Mar 06,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The value of the expression is -13.

Solución

Let's solve the problem step by step. ### Step 1. Simplify the Expression We are given the expression \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta}. \] Since the denominators are the same, we can combine the numerators: \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = \frac{10 - 5}{\cos \theta} = \frac{5}{\cos \theta}. \] ### Step 2. Determine \(\cos \theta\) from \(\tan \theta\) We are given \[ \tan \theta = \frac{12}{5}. \] A triangle corresponding to this relation has: - Opposite side \(= 12\) - Adjacent side \(= 5\) The hypotenuse of the triangle is \[ \text{hypotenuse} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13. \] Thus, if \(\theta\) were in the first quadrant, \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13}. \] However, we are given that \(90^\circ < \theta < 270^\circ\). In this interval, the angle is either in the second or third quadrant. Since \(\tan \theta = \frac{12}{5}\) is positive, \(\theta\) must be in the **third quadrant** (because tangent is positive in the first and third quadrants, and the first quadrant is excluded by the interval). In the third quadrant, the cosine is negative. Therefore, \[ \cos \theta = -\frac{5}{13}. \] ### Step 3. Substitute \(\cos \theta\) into the Expression Now substitute the value of \(\cos \theta\) into the simplified expression: \[ \frac{5}{\cos \theta} = \frac{5}{-\frac{5}{13}}. \] Simplify the fraction: \[ \frac{5}{-\frac{5}{13}} = 5 \times \left(-\frac{13}{5}\right) = -13. \] ### Final Answer The value of the given expression is: \[ \boxed{-13}. \]

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Bonus Knowledge

Let's start with the given information: \( \tan \theta = \frac{12}{5} \). We need to find \( \frac{10}{\cos \theta} - \frac{5}{\cos \theta} \). First, we can simplify the expression: \[ \frac{10}{\cos \theta} - \frac{5}{\cos \theta} = \frac{10 - 5}{\cos \theta} = \frac{5}{\cos \theta} \] Next, we need to determine the cosine of \( \theta \). Since \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5} \), we can represent the opposite and adjacent sides of a right triangle. The opposite side is 12, and the adjacent side is 5. Using the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Now we can find \( \cos \theta \): \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13} \] Now we substitute this value into our earlier expression: \[ \frac{5}{\cos \theta} = \frac{5}{\frac{5}{13}} = 5 \times \frac{13}{5} = 13 \] Thus, the value of \( \frac{10}{\cos \theta} - \frac{5}{\cos \theta} \) is \( \boxed{13} \).

preguntas relacionadas

Latest Trigonometry Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad