Question

Cos⁴x-sin⁴x=1/8 Найти:Sin²x

Answer 1

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Answer 2

To find the value of sin²x given the equation cos⁴x - sin⁴x = 1/8, we can use the trigonometric identity:

cos²x - sin²x = 1

Let's rewrite the equation cos⁴x - sin⁴x = 1/8 in terms of cos²x and sin²x:

(cos²x)² - (sin²x)² = 1/8

Using the identity cos²x - sin²x = 1, we can substitute it into the equation:

(1 - sin²x)² - sin²x = 1/8

Expanding the equation:

1 - 2sin²x + (sin²x)² - sin²x = 1/8

Simplifying:

1 - sin²x - sin²x + (sin⁴x) - sin²x = 1/8

Combining like terms:

1 - 3sin²x + sin⁴x = 1/8

Rearranging the equation:

sin⁴x - 3sin²x + 1 - 1/8 = 0

Now, we have a quadratic equation in terms of sin²x. Let's solve it:

sin⁴x - 3sin²x + 7/8 = 0

To simplify the equation, let's substitute y = sin²x:

y² - 3y + 7/8 = 0

To solve this quadratic equation, we can use the quadratic formula:

y = (-b ± √(b² - 4ac)) / (2a)

For this equation, a = 1, b = -3, and c = 7/8.

Plugging in these values, we have:

y = (-(-3) ± √((-3)² - 4(1)(7/8))) / (2(1))

Simplifying further:

y = (3 ± √(9 - 28/8)) / 2

y = (3 ± √(72/8 - 28/8)) / 2

y = (3 ± √(44/8)) / 2

y = (3 ± √(11/2)) / 2

Since sin²x must be between 0 and 1, we can discard the positive solution because it will be greater than 1. Therefore, we choose the negative solution:

y = (3 - √(11/2)) / 2

Now, substituting y = sin²x back into the equation:

sin²x = (3 - √(11/2)) / 2

Hence, the value of sin²x is (3 - √(11/2)) / 2.