The Function $h$ Is Given By $h(x) = 2 \sin^2 X + \sin X$. What Are The Zeros Of $ H H H [/tex] On The Interval $0 \leq X \ \textless \ 2\pi$? A. X = 0 X = 0 X = 0 B. X = Π X = \pi X = Π
Introduction
In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function h(x) = 2 sin^2 x + sin x is a trigonometric function that involves the sine of an angle x. In this article, we will explore the zeros of the function h(x) on the interval 0 ≤ x < 2π.
Understanding the Function h(x)
The function h(x) = 2 sin^2 x + sin x is a quadratic function in terms of sin x. To find the zeros of h(x), we need to find the values of x that make h(x) equal to zero. This means that we need to solve the equation 2 sin^2 x + sin x = 0.
Solving the Equation 2 sin^2 x + sin x = 0
To solve the equation 2 sin^2 x + sin x = 0, we can use the fact that sin x is a linear function in terms of sin^2 x. We can rewrite the equation as sin x(2 sin x + 1) = 0.
Factoring the Equation
The equation sin x(2 sin x + 1) = 0 can be factored as sin x = 0 or 2 sin x + 1 = 0.
Solving sin x = 0
The equation sin x = 0 has solutions x = 0, π, and 2π. However, since we are only interested in the interval 0 ≤ x < 2π, we can ignore the solution x = 2π.
Solving 2 sin x + 1 = 0
The equation 2 sin x + 1 = 0 can be rewritten as sin x = -1/2. This equation has solutions x = 7π/6 and 11π/6. However, since we are only interested in the interval 0 ≤ x < 2π, we can ignore the solution x = 11π/6.
Conclusion
In conclusion, the zeros of the function h(x) = 2 sin^2 x + sin x on the interval 0 ≤ x < 2π are x = 0 and x = 7π/6.
Discussion
The function h(x) = 2 sin^2 x + sin x is a quadratic function in terms of sin x. The zeros of h(x) are the values of x that make h(x) equal to zero. In this article, we have shown that the zeros of h(x) on the interval 0 ≤ x < 2π are x = 0 and x = 7π/6.
Final Answer
The final answer is x = 0 and x = 7π/6.
Additional Information
- The function h(x) = 2 sin^2 x + sin x is a quadratic function in terms of sin x.
- The zeros of h(x) are the values of x that make h(x) equal to zero.
- The interval 0 ≤ x < 2π is a closed interval that includes the values x = 0 and x = 2π.
- The function h(x) = 2 sin^2 x + sin x has a period of 2π, which means that the function repeats itself every 2π units of x.
References
- [1] "Trigonometry" by Michael Corral, 2015.
- [2] "Calculus" by Michael Spivak, 2008.
- [3] "Mathematics for Computer Science" by Eric Lehman, 2018.
Related Topics
- Trigonometry
- Quadratic functions
- Zeros of a function
- Periodic functions
Tags
- Trigonometry
- Quadratic functions
- Zeros of a function
- Periodic functions
- Mathematics
- Calculus
- Computer science
Introduction
In our previous article, we explored the function h(x) = 2 sin^2 x + sin x and its zeros on the interval 0 ≤ x < 2π. In this article, we will answer some frequently asked questions about the function h(x) and its zeros.
Q: What is the function h(x) = 2 sin^2 x + sin x?
A: The function h(x) = 2 sin^2 x + sin x is a quadratic function in terms of sin x. It is a trigonometric function that involves the sine of an angle x.
Q: What are the zeros of h(x) on the interval 0 ≤ x < 2π?
A: The zeros of h(x) on the interval 0 ≤ x < 2π are x = 0 and x = 7π/6.
Q: How do you find the zeros of a function?
A: To find the zeros of a function, you need to solve the equation f(x) = 0, where f(x) is the function.
Q: What is the difference between a zero and a root of a function?
A: A zero of a function is a value of x that makes the function equal to zero. A root of a function is a value of x that makes the function equal to zero, but it can also be a value of x that makes the function equal to a non-zero constant.
Q: Can you give an example of a function with no zeros?
A: Yes, the function f(x) = x^2 + 1 has no zeros, because there is no value of x that makes f(x) equal to zero.
Q: Can you give an example of a function with an infinite number of zeros?
A: Yes, the function f(x) = x(x - 1)(x - 2)... has an infinite number of zeros, because there are an infinite number of values of x that make f(x) equal to zero.
Q: How do you find the zeros of a quadratic function?
A: To find the zeros of a quadratic function, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic function.
Q: Can you give an example of a quadratic function with two zeros?
A: Yes, the function f(x) = x^2 - 4x + 4 has two zeros, x = 2 and x = 2.
Q: Can you give an example of a quadratic function with no zeros?
A: Yes, the function f(x) = x^2 + 1 has no zeros, because there is no value of x that makes f(x) equal to zero.
Q: How do you find the zeros of a periodic function?
A: To find the zeros of a periodic function, you need to find the values of x that make the function equal to zero, and then find the period of the function.
Q: Can you give an example of a periodic function with an infinite number of zeros?
A: Yes, the function f(x) = sin(x) has an infinite number of zeros, because there are an infinite number of values of x that make f(x) equal to zero.
Q: Can you give an example of a periodic function with no zeros?
A: Yes, the function f(x) = cos(x) has no zeros, because there is no value of x that makes f(x) equal to zero.
Conclusion
In this article, we have answered some frequently asked questions about the function h(x) = 2 sin^2 x + sin x and its zeros on the interval 0 ≤ x < 2π. We hope that this article has been helpful in understanding the function h(x) and its zeros.
Final Answer
The final answer is x = 0 and x = 7π/6.
Additional Information
- The function h(x) = 2 sin^2 x + sin x is a quadratic function in terms of sin x.
- The zeros of h(x) are the values of x that make h(x) equal to zero.
- The interval 0 ≤ x < 2π is a closed interval that includes the values x = 0 and x = 2π.
- The function h(x) = 2 sin^2 x + sin x has a period of 2π, which means that the function repeats itself every 2π units of x.
References
- [1] "Trigonometry" by Michael Corral, 2015.
- [2] "Calculus" by Michael Spivak, 2008.
- [3] "Mathematics for Computer Science" by Eric Lehman, 2018.
Related Topics
- Trigonometry
- Quadratic functions
- Zeros of a function
- Periodic functions
- Mathematics
- Calculus
- Computer science
Tags
- Trigonometry
- Quadratic functions
- Zeros of a function
- Periodic functions
- Mathematics
- Calculus
- Computer science