1. Let's start by understanding what a period means in the context of functions, especially trigonometric functions like sine and cosine. 2. The period of a function is the length of the interval over which the function completes one full cycle and then repeats. 3. You mentioned a full cycle is 10, but the period is 8. This suggests there might be a misunderstanding or a specific function where the cycle length and period differ. 4. For example, consider a function like $f(x) = \sin\left(\frac{2\pi}{8}x\right)$. 5. The period $T$ of a sine function $\sin(bx)$ is given by the formula: $$T = \frac{2\pi}{b}$$ 6. If $b = \frac{2\pi}{8}$, then the period is: $$T = \frac{2\pi}{\frac{2\pi}{8}} = 8$$ 7. Now, if someone says the "full cycle" is 10, they might be referring to the length of the domain interval they are observing, but the function itself repeats every 8 units. 8. So, the period is a property of the function's formula, while the observed cycle length might be different due to the domain or scaling. 9. In summary, the period is 8 because the function repeats every 8 units according to its formula, even if you observe a full cycle over 10 units in some context. Final answer: The period is 8 because it is determined by the function's formula and represents the length of one complete cycle, regardless of any other interval length like 10.