Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer space
below.

Ask by Burton Sullivan. in the United States

Mar 27,2025

Question

Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer space
below.

Ask by Burton Sullivan. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

We start with the inequality $$ 2\cos^2 \theta - \cos \theta - 1 > 0. $$ Let $$ x = \cos \theta $$ so that the inequality becomes $$ 2x^2 - x - 1 > 0. $$ 1. Solve the corresponding quadratic equation: $$ 2x^2 - x - 1 = 0. $$ Using the quadratic formula, where $$ a = 2 $$, $$ b = -1 $$, and $$ c = -1 $$: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm 3}{4}. $$ Thus, the solutions are $$ x = \frac{1+3}{4} = 1 \quad \text{and} \quad x = \frac{1-3}{4} = -\frac{1}{2}. $$ 2. Factor the quadratic: $$ 2x^2 - x - 1 = 2\left(x-1\right)\left(x+\frac{1}{2}\right). $$ Since $$ 2 > 0 $$, the inequality $$ 2\left(x-1\right)\left(x+\frac{1}{2}\right) > 0 $$ is equivalent to $$ \left(x-1\right)\left(x+\frac{1}{2}\right) > 0. $$ 3. Determine the intervals for which the product is positive: - The product is positive when both factors are positive: $$ x-1 > 0 \quad \text{and} \quad x+\frac{1}{2} > 0 \quad \Longrightarrow \quad x > 1. $$ However, since $$ \cos \theta \leq 1 $$, there are no solutions in this case. - The product is also positive when both factors are negative: $$ x-1 < 0 \quad \text{and} \quad x+\frac{1}{2} < 0 \quad \Longrightarrow \quad x < 1 \quad \text{and} \quad x < -\frac{1}{2}. $$ This simplifies to $$ x < -\frac{1}{2}. $$ 4. Translate back to $$ \cos \theta $$: We require $$ \cos \theta < -\frac{1}{2}. $$ On the interval $$ 0 \leq \theta \leq 2\pi $$, $$ \cos \theta = -\frac{1}{2} $$ at $$ \theta = \frac{2\pi}{3} $$ and $$ \theta = \frac{4\pi}{3} $$. The cosine function is less than $$-\frac{1}{2}$$ when $$ \theta \in \left(\frac{2\pi}{3}, \frac{4\pi}{3}\right). $$ Thus, the solution in interval notation is $$ \left(\frac{2\pi}{3}, \frac{4\pi}{3}\right). $$ The solution is $$ \theta $$ in the interval $$ \left(\frac{2\pi}{3}, \frac{4\pi}{3}\right) $$.

Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer space
below.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Trigonometry Questions

Question Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer space below.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Trigonometry Questions

Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer space
below.