How to find amplitude and period of a cos function?
Understand the Problem
The question is asking how to determine the amplitude and period of a cosine function. The amplitude is the maximum value of the function, and the period is the length of one complete wave cycle. These values can typically be found from the equation of the cosine function, which may be in the form y = A * cos(Bx), where |A| gives the amplitude and 2π/|B| gives the period.
Answer
Amplitude: $3$, Period: $\pi$.
Answer for screen readers
The amplitude is $3$, and the period is $\pi$.
Steps to Solve
- Identifying the Equation
First, write down the provided cosine function in the standard form $y = A \cdot \cos(Bx)$.
- Finding the Amplitude
The amplitude $A$ is the coefficient in front of the cosine function. To find the amplitude, take the absolute value of $A$.
- Calculating the Period
The period of the cosine function can be calculated using the formula $\text{Period} = \frac{2\pi}{|B|}$. Identify the value of $B$ from the equation and substitute it into the formula.
- Example Calculation
If the cosine function is $y = 3 \cdot \cos(2x)$, then:
- Amplitude is $|A| = |3| = 3$.
- To find the period, calculate $\frac{2\pi}{|B|}$, where $B = 2$: $$\text{Period} = \frac{2\pi}{2} = \pi$$
- Final Values
Conclude by stating both the amplitude and period clearly with the calculated values.
The amplitude is $3$, and the period is $\pi$.
More Information
This solution shows how the cosine function's amplitude and period can be easily identified from the function's equation. The amplitude informs us of the maximum height of the wave, while the period gives us the length of one complete cycle.
Tips
- Forgetting to take the absolute value of $A$ when determining amplitude.
- Confusing the formula for period with other trigonometric functions; remember the period formula specifically for cosine is $\frac{2\pi}{|B|}$.
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