Let $f(x) = \sin X$. What Are All Values Of $x$ In The $ X Y Xy X Y [/tex]-plane, $0 \leq X \leq 2\pi$, For Which $f(x) \geq -\frac{\sqrt{2}}{2}$?

Solving the Inequality: Values of x for f(x) ≥ -√2/2

In this article, we will explore the values of x in the xy-plane for which the function f(x) = sin(x) is greater than or equal to -√2/2. This inequality will be solved using mathematical techniques and graphical analysis.

Understanding the Function

The function f(x) = sin(x) is a periodic function that oscillates between -1 and 1. The sine function has a period of 2π, meaning that it repeats itself every 2π units of x.

Graphical Analysis

To visualize the inequality f(x) ≥ -√2/2, we can graph the function f(x) = sin(x) and the line y = -√2/2 on the same coordinate plane.

**Graph of f(x) = sin(x)**
-------------------------



x
sin(x)




0
0


π/6
1/2


π/4
√2/2


π/3
√3/2


π/2
1


2π/3
√3/2


5π/6
√2/2


3π/4
1/2


π
0


5π/4
-1/2


7π/4
-√2/2


4π/3
-√3/2


5π/2
-1


7π/3
-√3/2


11π/6
-√2/2


3π/2
-1


13π/6
-√2/2


5π/4
-1/2


3π
0


7π/4
1/2


11π/4
√2/2


4π/3
√3/2


3π/2
1


5π/3
√3/2


7π/4
√2/2


3π/2
1


11π/6
√2/2


2π
0

Solving the Inequality

To solve the inequality f(x) ≥ -√2/2, we need to find the values of x for which sin(x) ≥ -√2/2.

**Solving the Inequality**
-------------------------



x
sin(x)




0
0


π/6
1/2


π/4
√2/2


π/3
√3/2


π/2
1


2π/3
√3/2


5π/6
√2/2


3π/4
1/2


π
0


5π/4
-1/2


7π/4
-√2/2


4π/3
-√3/2


5π/2
-1


7π/3
-√3/2


11π/6
-√2/2


3π/2
-1


13π/6
-√2/2


5π/4
-1/2


3π
0


7π/4
1/2


11π/4
√2/2


4π/3
√3/2


3π/2
1


5π/3
√3/2


7π/4
√2/2


3π/2
1


11π/6
√2/2


2π
0

In conclusion, the values of x for which f(x) ≥ -√2/2 are:

• 0 ≤ x ≤ π/2
• 5π/6 ≤ x ≤ 3π/2
• 7π/4 ≤ x ≤ 2π

These values of x satisfy the inequality f(x) ≥ -√2/2.
Q&A: Solving the Inequality f(x) ≥ -√2/2

In our previous article, we explored the values of x in the xy-plane for which the function f(x) = sin(x) is greater than or equal to -√2/2. In this article, we will answer some frequently asked questions related to solving the inequality f(x) ≥ -√2/2.

Q: What is the significance of the value -√2/2 in the inequality f(x) ≥ -√2/2?

A: The value -√2/2 is a critical value in the sine function. It represents the minimum value of the sine function, which is -1. The inequality f(x) ≥ -√2/2 is satisfied when the sine function is greater than or equal to -√2/2.

Q: How do I graph the inequality f(x) ≥ -√2/2?

A: To graph the inequality f(x) ≥ -√2/2, you can graph the function f(x) = sin(x) and the line y = -√2/2 on the same coordinate plane. The region above the line y = -√2/2 represents the values of x for which the inequality is satisfied.

Q: What are the intervals of x for which the inequality f(x) ≥ -√2/2 is satisfied?

A: The intervals of x for which the inequality f(x) ≥ -√2/2 is satisfied are:

• 0 ≤ x ≤ π/2
• 5π/6 ≤ x ≤ 3π/2
• 7π/4 ≤ x ≤ 2π

Q: How do I determine the values of x for which the inequality f(x) ≥ -√2/2 is satisfied?

A: To determine the values of x for which the inequality f(x) ≥ -√2/2 is satisfied, you can use the following steps:

1. Graph the function f(x) = sin(x) and the line y = -√2/2 on the same coordinate plane.
2. Identify the region above the line y = -√2/2.
3. Determine the intervals of x for which the region above the line y = -√2/2 is satisfied.

Q: What is the relationship between the sine function and the inequality f(x) ≥ -√2/2?

A: The sine function is a periodic function that oscillates between -1 and 1. The inequality f(x) ≥ -√2/2 is satisfied when the sine function is greater than or equal to -√2/2. This means that the sine function must be in the range [0, 1] or [-1, -√2/2] to satisfy the inequality.

Q: Can I use other mathematical functions to solve the inequality f(x) ≥ -√2/2?

A: Yes, you can use other mathematical functions to solve the inequality f(x) ≥ -√2/2. For example, you can use the cosine function or the tangent function to solve the inequality. However, the sine function is the most straightforward function to use in this case.

In conclusion, solving the inequality f(x) ≥ -√2/2 requires a thorough understanding of the sine function and its behavior. By graphing the function and identifying the intervals of x for which the inequality is satisfied, you can determine the values of x for which the inequality is true.