Question

3sin^2x+7sinx×cosx=5​

Answer 1

Ответ:

x1 = arctg(5/2) + пk, (k - целое)

x2 = п/4 + пn, (n - целое)

Пошаговое объяснение:

3*(sin(x))^2 + 7*sin(x)cos(x) = 5

3*sin(x)^2 + 7*sin(x)cos(x) = 5(sin(x)^2 + cos(x)^2)

0 = 2*sin(x)^2 - 7*sin(x)cos(x) + 5*cos(x)^2    || : cos(x)^2

0 = 2*tg(x)^2 - 7*tg(x) + 5

Пусть  tg(x) = t

2t^2 - 7t + 5 = 0

D = 49 - 4*2*5 = 49 -40 = 9

t1 = (7 + 3)/(2*2) = 10/4 = 2.5

t2 = (7 - 3)/(2*2) = 4/4 = 1

x1 = arctg(5/2) + пk, (k - целое)

x2 = arctg(1) + пn = п/4 + пn, (n - целое)

Answer 2

To solve the equation $3\sin^2(x) + 7\sin(x)\cos(x) = 5$, we can use trigonometric identities to simplify and solve for $\sin(x)$ and $\cos(x)$.

First, note that $2\sin(x)\cos(x) = \sin(2x)$ using the double angle formula for sine. So, we can rewrite the equation as follows:

$3\sin^2(x) + 7\sin(x)\cos(x) = 5$ $3\sin^2(x) + 7\sin(2x) = 5$

Now, we can use the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$ to express $\sin^2(x)$ in terms of $\cos(x)$:

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

Substituting this into the equation:

$3(1 - \cos^2(x)) + 7\sin(2x) = 5$

Next, we can use the double angle formula for sine $\sin(2x) = 2\sin(x)\cos(x)$ to replace $\sin(2x)$:

$3(1 - \cos^2(x)) + 14\sin(x)\cos(x) = 5$

Now, we can substitute $2\sin(x)\cos(x)$ with $\sin(2x)$ again:

$3(1 - \cos^2(x)) + 14\sin(2x) = 5$

Now, the equation is solely in terms of $\cos(x)$ and $\sin(2x)$. Let's focus on $\sin(2x)$:

$14\sin(2x) = 5 - 3(1 - \cos^2(x))$

$14\sin(2x) = 5 - 3 + 3\cos^2(x)$

Now, isolate $\cos^2(x)$:

$3\cos^2(x) = 14\sin(2x) - 2$

$\cos^2(x) = \frac{14\sin(2x) - 2}{3}$

Now, recall the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$. We can use it to find $\sin(2x)$:

$\sin^2(2x) + \cos^2(2x) = 1$

Since $\cos(2x) = \cos^2(x) - \sin^2(x)$, we can substitute it:

$\sin^2(2x) + (\cos^2(x) - \sin^2(x)) = 1$

Now, use the identity $\sin(2x) = 2\sin(x)\cos(x)$ for $\sin(2x)$:

$4\sin(x)\cos(x) + (\cos^2(x) - \sin^2(x)) = 1$

Now, we have two equations for $\cos^2(x)$:

1. $\cos^2(x) = \frac{14\sin(2x) - 2}{3}$
2. $\cos^2(x) - \sin^2(x) + 4\sin(x)\cos(x) = 1$

You can solve these two equations simultaneously to find the values of $\cos(x)$ and $\sin(x)$. Once you have those values, you can find $x$ using the inverse trigonometric functions, such as $\arcsin$ and $\arccos$. Note that these equations can be quite complex to solve analytically, and you may need to use numerical methods or a calculator to approximate the solutions.