Question

Решить уравнения:cos^(-2) 2t-sin^(-2) 2t=8/3

Answer 1

Решение этого уравнения - во вложении

Answer 2

To solve the equation cos^(-2)(2t) - sin^(-2)(2t) = 8/3, we'll first manipulate the trigonometric expressions to make it easier to work with. The following trigonometric identity will be useful:

1 - sin^2(x) = cos^2(x)

Using this identity, we can rewrite the equation:

cos^(-2)(2t) - sin^(-2)(2t) = cos^2(2t) - sin^2(2t) = 8/3

Now, we can use another trigonometric identity:

cos(2x) = cos^2(x) - sin^2(x)

Substitute 2t for x:

cos(2(2t)) = cos^2(2t) - sin^2(2t)

cos(4t) = cos^2(2t) - sin^2(2t)

Now, let's rewrite the original equation with the new identity:

cos(4t) = 8/3

To solve for t, we'll take the inverse cosine (also known as arccos) of both sides:

arccos(cos(4t)) = arccos(8/3)

The arccos and cos functions are inverses of each other, so they cancel out, and we get:

4t = arccos(8/3)

Now, divide both sides by 4 to isolate t:

t = (1/4) * arccos(8/3)

The above expression is the exact solution for t. If you want a numerical approximation, you can use a calculator to evaluate arccos(8/3) and then multiply by 1/4.