what is $$ 8\cdot\left(\sin x\right) $$^4

To simplify and understand the expression $ 8 \cdot \left(\sin x\right)^4 $, we recognize that it represents eight times the fourth power of the sine of $ x $. Mathematically, it can be written as:

\[ 8 \left(\sin x\right)^4 \]

Here are a few observations about this expression:

1. **Basic Properties**: $ \sin x $ ranges from -1 to 1 for all $ x $. Thus, $ \left(\sin x\right)^4 $ ranges from 0 to 1 because the fourth power of any real number is non-negative.
2. **Periodicity**: Since $ \sin x $ is periodic with a period of $ 2\pi $, $ \left(\sin x\right)^4 $ is also periodic with a period of $ 2\pi $.
3. **Maxima and Minima**: The expression achieves its maximum value of 8 when $ \sin x = \pm 1 $ (since $(\pm 1)^4 = 1$), resulting in $ 8 \cdot 1 = 8 $. Its minimum value is 0, which occurs when $ \sin x = 0 $.

Would you like further details or have any specific questions about this expression?

Here are some relative questions that expand on this information:

1. How would you graph the function $ f(x) = 8 \left(\sin x\right)^4 $?
2. What is the derivative of $ 8 \left(\sin x\right)^4 $ with respect to $ x $?
3. How does the expression $ 8 \left(\sin x\right)^4 $ change if we replace $ x $ with $ 2x $?
4. What are the critical points of the function $ f(x) = 8 \left(\sin x\right)^4 $?
5. Can you express $ 8 \left(\sin x\right)^4 $ in terms of cosine using trigonometric identities?
6. What is the integral of $ 8 \left(\sin x\right)^4 $ with respect to $ x $?
7. How does the expression $ 8 \left(\sin x\right)^4 $ behave as $ x $ approaches infinity?
8. What is the Taylor series expansion of $ \left(\sin x\right)^4 $ around $ x = 0 $?

**Tip:** When dealing with trigonometric functions raised to a power, using trigonometric identities can simplify the expression and make calculus operations easier.

Explore the mathematical expression 8 times the fourth power of the sine of x (8(sin x)^4) in trigonometry. Learn about its properties, including its range, periodicity, and critical points. Suitable for advanced high school students.

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