Question

Помогите,СРОЧНО,пожалуйста Найти область значений функции у=1/(sin^4(x)+cos^4(x))

Answer 1

$sin ^{4} x+cos ^{4} x=sin ^{4} x+cos ^{4} x-2sin ^{2} x*cos ^{2} x+2sin ^{2} x*cos ^{2} x= \\ (sin ^{4} x+2sin ^{2} x*cos ^{2} x+cos ^{4} x)-2sin ^{2} x*cos ^{2} x= \\ (sin ^{2} x+cos ^{2} x) ^{2} -2sin ^{2} x*cos ^{2} x=[tex]y= \frac{1}{sin ^{4} x+cos ^{4} x} \ \textless \ =\ \textgreater \ y= \frac{1}{1- \frac{1}{2} sin ^{2} (2x)} \\ 0\leq sin ^{2} (2x) \leq 1 \\ - \frac{1}{2} \leq - \frac{1}{2} sin ^{2}(2x) \leq 0 \\ \frac{1}{2} \leq 1- \frac{1}{2} sin ^{2} (2x) \leq 1 \\1 \leq \frac{1}{1- \frac{1}{2} sin ^{2} (2x)} \leq 2 \\ E(y)=[1;2]$[/tex]

Answer 2

Finding the Range of the Function y=1/(sin^4(x)+cos^4(x))

To find the range of the function y=1/(sin^4(x)+cos^4(x)), we need to consider the properties of the given function and its components.

The function y=1/(sin^4(x)+cos^4(x)) involves trigonometric functions and their powers. The range of this function will depend on the values that sin^4(x) and cos^4(x) can take.

Trigonometric Identities

Trigonometric identities can help us simplify the expression and understand the behavior of sin^4(x) and cos^4(x). One such identity is: cos^4(x) + 2cos^2(x)sin^2(x) + sin^4(x) = 1

This identity can be used to simplify the expression sin^4(x) + cos^4(x) in terms of sin^2(x) and cos^2(x).

Simplifying the Expression

Using the trigonometric identity, we can rewrite sin^4(x) + cos^4(x) as: sin^4(x) + cos^4(x) = 1 - 2cos^2(x)sin^2(x)

This allows us to express y=1/(1 - 2cos^2(x)sin^2(x)).

Analyzing the Range

The expression 1 - 2cos^2(x)sin^2(x) can take values between 0 and 1. Therefore, the range of the function y=1/(sin^4(x)+cos^4(x)) will be determined by the range of 1 - 2cos^2(x)sin^2(x).

Conclusion

The range of the function y=1/(sin^4(x)+cos^4(x)) is determined by the range of 1 - 2cos^2(x)sin^2(x), which is between 0 and 1.

Note: Further analysis or specific values for the range may require additional mathematical techniques or software tools.

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