Question

Sin³a+cos³a если sina+cosa=1/3

Answer 1

Ответ:

$\frac{13}{27}$

Объяснение:

Answer 2

We can use the given trigonometric identity to find the value of $\sin^3(a) + \cos^3(a)$ when $\sin(a) + \cos(a) = \frac{1}{3}$.

The identity we'll use is:

$\sin^3(a) + \cos^3(a) = (\sin(a) + \cos(a))(1 - \sin(a)\cos(a))$

Given that $\sin(a) + \cos(a) = \frac{1}{3}$, we substitute this value into the identity:

$\sin^3(a) + \cos^3(a) = \left(\frac{1}{3}\right) \left(1 - \sin(a)\cos(a)\right)$

Now we need to find $\sin(a)\cos(a)$. We're given that $\sin(a) + \cos(a) = \frac{1}{3}$, so we can square this equation to get:

$\sin^2(a) + 2\sin(a)\cos(a) + \cos^2(a) = \frac{1}{9}$

Since $\sin^2(a) + \cos^2(a) = 1$, we can substitute this into the equation:

$1 + 2\sin(a)\cos(a) = \frac{1}{9}$

Solving for $\sin(a)\cos(a)$:

$2\sin(a)\cos(a) = \frac{1}{9} - 1$ $2\sin(a)\cos(a) = -\frac{8}{9}$ $\sin(a)\cos(a) = -\frac{4}{9}$

Now we can substitute this value back into the original identity:

$\sin^3(a) + \cos^3(a) = \left(\frac{1}{3}\right) \left(1 - \left(-\frac{4}{9}\right)\right)$ $\sin^3(a) + \cos^3(a) = \frac{1}{3} \left(\frac{13}{9}\right)$ $\sin^3(a) + \cos^3(a) = \frac{13}{27}$

So, $\sin^3(a) + \cos^3(a) = \frac{13}{27}$ when $\sin(a) + \cos(a) = \frac{1}{3}$.