- The frequency \(f\) is the reciprocal of the period \(P\), resulting in \(f = \frac{1}{1000}\).
- Using the relation \(P = \frac{360^\circ}{b}\), the coefficient \(b\) is calculated as \(b = \frac{360}{1000} = 0.36\).
- None of the provided equations perfectly match a period of \(1000^\circ\), but the derivation establishes that the correct coefficient is \(\boxed{0.36}\).
Explanation
- Calculating the frequency and the coefficient \(b\)
The period of a periodic function is the length of one complete cycle. For a cosine function of the form \(y = \cos(bx)\) or \(y = A \cos(bx - c) + k\), the period \(P\) is related to the coefficient \(b\) by the formula \(P = \frac{360^\circ}{|b|}\) (when the period is measured in degrees). Given that the period of our function is \(P = 1000^\circ\), we can determine the value of \(b\) by rearranging the formula: \[b = \frac{360^\circ}{P} = \frac{360^\circ}{1000^\circ} = 0.36.\] The frequency \(f\) of a function is defined as the number of cycles per unit interval, which is the reciprocal of the period: \[f = \frac{1}{P} = \frac{1}{1000} = 0.001.\]
- Evaluating the provided equations
We are given three possible equations for our cosine function:
- \(y = \cos(x - 3)\): The coefficient of \(x\) is \(b = 1\), so the period is \(P = \frac{360^\circ}{1} = 360^\circ\).
- \(y = \cos(x / 3)\): The coefficient of \(x\) is \(b = \frac{1}{3}\), so the period is \(P = \frac{360^\circ}{1/3} = 360^\circ \times 3 = 1080^\circ\).
- \(y = \cos(3x)\): The coefficient of \(x\) is \(b = 3\), so the period is \(P = \frac{360^\circ}{3} = 120^\circ\).
Comparing these calculated periods to our given period of \(1000^\circ\), we observe that none of the provided equations exactly match a period of \(1000^\circ\). However, in educational contexts where such a question might appear with potential typos, the equation \(y = \cos(x / 3)\) (with a period of \(1080^\circ\)) is the closest approximation to the target period of \(1000^\circ\). Based on the formal mathematical derivation, the correct coefficient for a period of exactly \(1000^\circ\) would be \(b = 0.36\), making the equation \(y = \cos(0.36x)\). Since that is not an option, we identify the values found through derivation.
- Final Conclusion
Based on our calculations:
- Frequency: \(f = \frac{1}{1000}\)
- Value of \(b\): \(b = 0.36\) (derived as \(\frac{360}{1000} = \frac{9}{25}\))
- Given the options, none produce a period of \(1000^\circ\). The closest match is \(y = \cos(x/3)\), although it yields a period of \(1080^\circ\). Assuming a potential error in the source material's provided options, the correct \(b\) is \(\boxed{0.36}\).
Examples
In music, the frequency of a sound wave determines its pitch. If you are designing an electronic synthesizer, you need to understand how the period of the sound wave (the length of time for one oscillation) translates into the frequency parameter \(b\) in the signal function \(y = A \cos(bx)\). Knowing that \(f = 1/P\) allows you to set the correct frequency to produce a specific note, just as we calculated the parameters for our cosine function here.