### Problem StatementWhat is the smallest positive solution to $5\sin^2 x - \cos^2 x = 1$? Round to 2 decimal places.---### ExplanationThis problem involves solving a trigonometric equation to find the smallest positive solution for $x$. The equation combines both sine and cosine functions, and solving it typically involves using trigonometric identities and algebraic manipulation.1. **Trigonometric Identity:** One key identity that might be useful is $\sin^2 x + \cos^2 x = 1$.2. **Algebraic Manipulation:** You may try expressing the equation in terms of a single trigonometric function, using the identity mentioned above to simplify $\cos^2 x$.3. **Potential Method of Solution:** - Substitute $\cos^2 x$ with $1 - \sin^2 x$. - Solve the resulting quadratic equation in terms of $\sin x$. - Find the smallest positive solution for $x$.This exercise is a practice of combining knowledge of trigonometry with algebraic techniques to find a specific numerical solution.

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Description: ### Problem StatementWhat is the smallest positive solution to \(5\sin^2 x - \cos^2 x = 1$? Round to 2 decimal places.---### ExplanationThis problem involves solving a trigonometric equation to find the smallest positive solution for $x$. The equ

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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### Problem Statement

What is the smallest positive solution to $5\sin^2 x - \cos^2 x = 1$? Round to 2 decimal places.

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### Explanation

This problem involves solving a trigonometric equation to find the smallest positive solution for $x$. The equation combines both sine and cosine functions, and solving it typically involves using trigonometric identities and algebraic manipulation.

1. **Trigonometric Identity:** One key identity that might be useful is $\sin^2 x + \cos^2 x = 1$.

2. **Algebraic Manipulation:** You may try expressing the equation in terms of a single trigonometric function, using the identity mentioned above to simplify $\cos^2 x$.

3. **Potential Method of Solution:**
- Substitute $\cos^2 x$ with $1 - \sin^2 x$.
- Solve the resulting quadratic equation in terms of $\sin x$.
- Find the smallest positive solution for $x$.

This exercise is a practice of combining knowledge of trigonometry with algebraic techniques to find a specific numerical solution.

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