Question

Y=(sin^3(x)+cos^3(x))/(sin^3(x)-cos^3(x)),если tg(x)=2

Answer 1

Смотри решение на фото.

При решении использовалось основное свойство дроби: числитель и знаменатель дроби можно разделить на одно и то же число, при этом дробь не изменится.

Answer 2

To find the value of Y when tg(x) = 2, we first need to determine the value of x.

We know that tg(x) = sin(x)/cos(x). If tg(x) = 2, then sin(x)/cos(x) = 2.

We can rearrange this equation as sin(x) = 2cos(x).

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite sin(x) as sqrt(1 - cos^2(x)). Substituting this into the equation above, we have:

sqrt(1 - cos^2(x)) = 2cos(x).

Squaring both sides of the equation, we get:

1 - cos^2(x) = 4cos^2(x).

Rearranging the equation, we have:

5cos^2(x) = 1.

Dividing both sides by 5, we obtain:

cos^2(x) = 1/5.

Taking the square root of both sides, we get:

cos(x) = ±sqrt(1/5).

Since we know that tg(x) = 2, and tg(x) = sin(x)/cos(x), we can calculate sin(x) as:

sin(x) = 2cos(x).

Substituting the value of cos(x), we have:

sin(x) = 2 * ±sqrt(1/5).

Therefore, there are two possible values for sin(x), which are ±2 * sqrt(1/5).

Now that we have the values of sin(x) and cos(x), we can calculate Y using the given formula:

Y = (sin^3(x) + cos^3(x))/(sin^3(x) - cos^3(x)).

Substituting the values we obtained, we have:

Y = ([±2 * sqrt(1/5)]^3 + [±sqrt(1/5)]^3)/([±2 * sqrt(1/5)]^3 - [±sqrt(1/5)]^3).

Simplifying further, we get:

Y = (8/5 ± 2sqrt(1/5) + 1/5)/(8/5 ± 2sqrt(1/5) - 1/5).

The final result for Y will depend on the specific values chosen for sin(x) and cos(x).