The Period of |sin(x)| and |cos(x)|

The period of a function is the length of the interval over which the function repeats its values. In this case, we are interested in finding the period of the absolute values of the sine and cosine functions, denoted as |sin(x)| and |cos(x)|, respectively.

Period of |sin(x)|

The sine function, sin(x), has a period of 2π, which means it repeats itself every 2π units along the x-axis. However, when we take the absolute value of sin(x), the negative values of sin(x) become positive, effectively resulting in a reflection of the negative portion of the sine curve across the x-axis.

Due to this reflection, the period of |sin(x)| is halved to π. This means that the absolute value of sin(x) repeats its values every π units along the x-axis. Therefore, the period of |sin(x)| is π.

Period of |cos(x)|

Similarly, the cosine function, cos(x), also has a period of 2π. However, when we take the absolute value of cos(x), the negative values of cos(x) become positive, resulting in a reflection of the negative portion of the cosine curve across the x-axis.

As a result of this reflection, the period of |cos(x)| remains the same as the original cosine function, which is 2π. The absolute value of cos(x) repeats its values every 2π units along the x-axis, just like cos(x) itself.

Comparison

To summarize, the period of |sin(x)| is π, while the period of |cos(x)| is 2π. This means that the absolute value of sin(x) repeats its values every π units, whereas the absolute value of cos(x) repeats its values every 2π units along the x-axis.

Key Points:
- The period of |sin(x)| is π, which means it repeats every π units along the x-axis.
- The period of |cos(x)| is 2π, which means it repeats every 2π units along the x-axis.
- The absolute value of sin(x) reflects the negative portion of the sine curve across the x-axis, halving its period.
- The absolute value of cos(x) also reflects the negative portion of the cosine curve across the x-axis, but it does not affect its period.