Let F ( X ) = 3 + 4 Sin ⁡ X F(x) = 3 + 4 \sin X F ( X ) = 3 + 4 Sin X And Let G ( X ) = 1 G(x) = 1 G ( X ) = 1 . What Are All Values Of X X X In The X Y Xy X Y -plane, 0 ≤ X ≤ 2 Π 0 \leq X \leq 2\pi 0 ≤ X ≤ 2 Π , For Which F ( X ) \textless G ( X F(x) \ \textless \ G(x F ( X ) \textless G ( X ]?

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Introduction

In this article, we will explore the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which the function $f(x) = 3 + 4 \sin x$ is less than the function $g(x) = 1$. This problem involves understanding the behavior of the sine function and its impact on the function $f(x)$.

Understanding the Sine Function

The sine function, denoted by $\sin x$, is a periodic function that oscillates between $-1$ and $1$. The graph of the sine function is a wave-like curve that repeats every $2\pi$ radians. The sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right triangle.

Analyzing the Function $f(x)$

The function $f(x) = 3 + 4 \sin x$ is a sinusoidal function that has been shifted upward by $3$ units and scaled by a factor of $4$. This means that the amplitude of the function $f(x)$ is $4$, and the vertical shift is $3$ units.

Finding the Values of $x$ for Which $f(x) \ \textless \ g(x)$

To find the values of $x$ for which $f(x) \ \textless \ g(x)$, we need to set up the inequality $3 + 4 \sin x < 1$ and solve for $x$. This inequality can be rewritten as $4 \sin x < -2$.

Solving the Inequality

To solve the inequality $4 \sin x < -2$, we need to isolate the sine function. Dividing both sides of the inequality by $4$, we get $\sin x < -\frac{1}{2}$.

Understanding the Behavior of the Sine Function

The sine function is negative in the third and fourth quadrants of the unit circle. Therefore, the inequality $\sin x < -\frac{1}{2}$ is satisfied when $x$ is in the third or fourth quadrant.

Finding the Values of $x$ in the Third and Fourth Quadrants

In the third quadrant, the sine function is negative and has a value greater than $-\frac{1}{2}$. Therefore, the inequality $\sin x < -\frac{1}{2}$ is satisfied when $x$ is in the third quadrant.

In the fourth quadrant, the sine function is negative and has a value less than $-\frac{1}{2}$. Therefore, the inequality $\sin x < -\frac{1}{2}$ is satisfied when $x$ is in the fourth quadrant.

Finding the Values of $x$ in the Fourth Quadrant

To find the values of $x$ in the fourth quadrant, we need to find the angles in the fourth quadrant that satisfy the inequality $\sin x < -\frac{1}{2}$. Using a calculator or a trigonometric table, we can find that the angles in the fourth quadrant that satisfy the inequality are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.

Finding the Values of $x$ in the Third Quadrant

To find the values of $x$ in the third quadrant, we need to find the angles in the third quadrant that satisfy the inequality $\sin x < -\frac{1}{2}$. Using a calculator or a trigonometric table, we can find that the angles in the third quadrant that satisfy the inequality are $\frac{4\pi}{3}$ and $\frac{5\pi}{3}$.

Conclusion

In conclusion, the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) \ \textless \ g(x)$ are $\frac{4\pi}{3}$, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, and $\frac{11\pi}{6}$.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Trigonometry

Discussion

This problem involves understanding the behavior of the sine function and its impact on the function $f(x)$. The values of $x$ for which $f(x) \ \textless \ g(x)$ are $\frac{4\pi}{3}$, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, and $\frac{11\pi}{6}$. This problem can be used to illustrate the importance of understanding the behavior of trigonometric functions in solving mathematical problems.

Related Problems

• [1] Find the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) = 3 + 4 \sin x$ is greater than $g(x) = 1$.
• [2] Find the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) = 3 + 4 \sin x$ is equal to $g(x) = 1$.

Tags

• Trigonometry
• Calculus
• Mathematics
• Sine function
• Inequality
• Quadrants
• Angles
• Trigonometric functions
  

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Q: What is the function $f(x)$ and what is its behavior?

A: The function $f(x)$ is defined as $f(x) = 3 + 4 \sin x$. This function is a sinusoidal function that has been shifted upward by $3$ units and scaled by a factor of $4$. The amplitude of the function $f(x)$ is $4$, and the vertical shift is $3$ units.

Q: What is the function $g(x)$ and what is its behavior?

A: The function $g(x)$ is defined as $g(x) = 1$. This function is a horizontal line that intersects the $y$-axis at $1$.

Q: How do we find the values of $x$ for which $f(x) \ \textless \ g(x)$?

A: To find the values of $x$ for which $f(x) \ \textless \ g(x)$, we need to set up the inequality $3 + 4 \sin x < 1$ and solve for $x$. This inequality can be rewritten as $4 \sin x < -2$.

Q: How do we solve the inequality $4 \sin x < -2$?

A: To solve the inequality $4 \sin x < -2$, we need to isolate the sine function. Dividing both sides of the inequality by $4$, we get $\sin x < -\frac{1}{2}$.

Q: What are the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) \ \textless \ g(x)$?

A: The values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) \ \textless \ g(x)$ are $\frac{4\pi}{3}$, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, and $\frac{11\pi}{6}$.

Q: Why are the values of $x$ in the third and fourth quadrants?

A: The values of $x$ in the third and fourth quadrants are because the sine function is negative in these quadrants. Therefore, the inequality $\sin x < -\frac{1}{2}$ is satisfied when $x$ is in the third or fourth quadrant.

Q: How do we find the values of $x$ in the fourth quadrant?

A: To find the values of $x$ in the fourth quadrant, we need to find the angles in the fourth quadrant that satisfy the inequality $\sin x < -\frac{1}{2}$. Using a calculator or a trigonometric table, we can find that the angles in the fourth quadrant that satisfy the inequality are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.

Q: How do we find the values of $x$ in the third quadrant?

A: To find the values of $x$ in the third quadrant, we need to find the angles in the third quadrant that satisfy the inequality $\sin x < -\frac{1}{2}$. Using a calculator or a trigonometric table, we can find that the angles in the third quadrant that satisfy the inequality are $\frac{4\pi}{3}$ and $\frac{5\pi}{3}$.

Q: What is the importance of understanding the behavior of trigonometric functions in solving mathematical problems?

A: Understanding the behavior of trigonometric functions is crucial in solving mathematical problems. Trigonometric functions are used to model real-world phenomena, and their behavior can have a significant impact on the solution of a problem.

Q: What are some related problems that can be used to illustrate the importance of understanding the behavior of trigonometric functions?

A: Some related problems that can be used to illustrate the importance of understanding the behavior of trigonometric functions are:

• Find the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) = 3 + 4 \sin x$ is greater than $g(x) = 1$.
• Find the values of $x$ in the $xy$-plane, $0 \leq x \leq 2\pi$, for which $f(x) = 3 + 4 \sin x$ is equal to $g(x) = 1$.

Q: What are some additional resources that can be used to learn more about trigonometric functions?

A: Some additional resources that can be used to learn more about trigonometric functions are:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Trigonometry

Q: What are some common mistakes that can be made when solving trigonometric problems?

A: Some common mistakes that can be made when solving trigonometric problems are:

• Not understanding the behavior of trigonometric functions
• Not using the correct trigonometric identities
• Not checking the units of the answer

Q: How can we avoid making these mistakes?

A: To avoid making these mistakes, we need to:

• Understand the behavior of trigonometric functions
• Use the correct trigonometric identities
• Check the units of the answer

Q: What are some tips for solving trigonometric problems?

A: Some tips for solving trigonometric problems are:

• Use a calculator or a trigonometric table to find the values of trigonometric functions
• Check the units of the answer
• Use the correct trigonometric identities

Q: What are some common applications of trigonometric functions?

A: Some common applications of trigonometric functions are:

• Modeling real-world phenomena
• Solving mathematical problems
• Calculating distances and angles in geometry and trigonometry

Q: What are some common areas where trigonometric functions are used?

A: Some common areas where trigonometric functions are used are:

• Physics
• Engineering
• Computer Science
• Mathematics

Q: What are some common tools that can be used to solve trigonometric problems?

A: Some common tools that can be used to solve trigonometric problems are:

• Calculators
• Trigonometric tables
• Graphing calculators
• Computer software

Q: What are some common resources that can be used to learn more about trigonometric functions?

A: Some common resources that can be used to learn more about trigonometric functions are:

• Textbooks
• Online resources
• Video lectures
• Practice problems