Which Of The Following Is True For F ( X ) = 5 Cos ⁡ ( X ) + 1 F(x) = 5 \cos(x) + 1 F ( X ) = 5 Cos ( X ) + 1 ?A. The Period Of The Function Is 10 Π 10\pi 10 Π .B. The Function Has An Amplitude Of 2.5.C. A Zero Of The Function Is \left(\frac{\pi}{2}, 0\right ].D. The Range Of The

Introduction

When dealing with trigonometric functions, it's essential to understand their properties, such as period, amplitude, zeros, and range. In this article, we'll explore the properties of the function $f(x) = 5 \cos(x) + 1$ and determine which of the given statements is true.

Period of the Function

The period of a trigonometric function is the distance along the x-axis over which the function repeats itself. For the function $f(x) = 5 \cos(x) + 1$, we need to find its period.

The period of the cosine function is $2\pi$. However, when the cosine function is multiplied by a constant, the period remains the same. Therefore, the period of the function $f(x) = 5 \cos(x) + 1$ is also $2\pi$.

Amplitude of the Function

The amplitude of a trigonometric function is the maximum value that the function attains. For the function $f(x) = 5 \cos(x) + 1$, we need to find its amplitude.

The amplitude of the cosine function is 1. However, when the cosine function is multiplied by a constant, the amplitude is also multiplied by the same constant. Therefore, the amplitude of the function $f(x) = 5 \cos(x) + 1$ is $5 \times 1 = 5$.

Zeros of the Function

A zero of a function is a value of x for which the function attains the value 0. For the function $f(x) = 5 \cos(x) + 1$, we need to find its zeros.

The zeros of the cosine function are given by the equation $\cos(x) = 0$. This equation has solutions $x = \frac{\pi}{2} + k\pi$, where k is an integer.

However, when the cosine function is multiplied by a constant, the zeros are also affected. In this case, the zeros of the function $f(x) = 5 \cos(x) + 1$ are given by the equation $5 \cos(x) + 1 = 0$. Solving this equation, we get $\cos(x) = -\frac{1}{5}$. This equation has solutions $x = \cos^{-1}(-\frac{1}{5}) + k\pi$, where k is an integer.

Range of the Function

The range of a function is the set of all possible values that the function attains. For the function $f(x) = 5 \cos(x) + 1$, we need to find its range.

The range of the cosine function is $[-1, 1]$. However, when the cosine function is multiplied by a constant and shifted by a constant, the range is also affected. In this case, the range of the function $f(x) = 5 \cos(x) + 1$ is given by $[1 - 5, 1 + 5] = [-4, 6]$.

Conclusion

In conclusion, we have analyzed the properties of the function $f(x) = 5 \cos(x) + 1$ and determined which of the given statements is true.

• The period of the function is $2\pi$, not $10\pi$.
• The amplitude of the function is 5, not 2.5.
• A zero of the function is not $\left(\frac{\pi}{2}, 0\right)$.
• The range of the function is $[-4, 6]$, not the entire real line.

Therefore, the correct answer is none of the above.

Introduction

In our previous article, we explored the properties of the function $f(x) = 5 \cos(x) + 1$. In this article, we'll answer some frequently asked questions about trigonometric functions.

Q: What is the period of a trigonometric function?

A: The period of a trigonometric function is the distance along the x-axis over which the function repeats itself. For the function $f(x) = 5 \cos(x) + 1$, the period is $2\pi$.

Q: How do I find the amplitude of a trigonometric function?

A: To find the amplitude of a trigonometric function, you need to find the maximum value that the function attains. For the function $f(x) = 5 \cos(x) + 1$, the amplitude is 5.

Q: What is a zero of a trigonometric function?

A: A zero of a trigonometric function is a value of x for which the function attains the value 0. For the function $f(x) = 5 \cos(x) + 1$, the zeros are given by the equation $5 \cos(x) + 1 = 0$.

Q: How do I find the range of a trigonometric function?

A: To find the range of a trigonometric function, you need to find the set of all possible values that the function attains. For the function $f(x) = 5 \cos(x) + 1$, the range is $[-4, 6]$.

Q: What is the difference between a sine and a cosine function?

A: The sine and cosine functions are both trigonometric functions, but they have different periods and amplitudes. The sine function has a period of $2\pi$ and an amplitude of 1, while the cosine function has a period of $2\pi$ and an amplitude of 1.

Q: Can I use trigonometric functions to model real-world phenomena?

A: Yes, trigonometric functions can be used to model many real-world phenomena, such as the motion of a pendulum, the vibration of a spring, and the oscillation of a circuit.

Q: How do I graph a trigonometric function?

A: To graph a trigonometric function, you need to plot the function's values for different values of x. You can use a graphing calculator or a computer program to graph the function.

Q: What are some common applications of trigonometric functions?

A: Trigonometric functions have many applications in science, engineering, and mathematics, including:

• Modeling the motion of objects
• Analyzing the behavior of electrical circuits
• Solving problems in physics and engineering
• Creating computer graphics and animations

Conclusion

In conclusion, we've answered some frequently asked questions about trigonometric functions. We hope this article has been helpful in understanding the properties and applications of trigonometric functions.

Additional Resources

If you're interested in learning more about trigonometric functions, we recommend the following resources:

• Online tutorials and videos
• Textbooks and reference books
• Online courses and degree programs
• Professional organizations and conferences

Final Thoughts

Trigonometric functions are an essential part of mathematics and science. They have many applications in real-world phenomena and are used to model and analyze complex systems. We hope this article has been helpful in understanding the properties and applications of trigonometric functions.