Question

4cosxsin(30+x)cos(60+x)=cos2 3x

Answer 1

Ответ:

Объяснение:

4cosxsin(30+x)cos(60+x)=cos^2(3x)

4cosx(1/4 cos^2(x) - 3/4 sin^2(x)) = cos^2(3x)

cos^3(x) - 3sin^2(x)cos(x) = cos^2(3x). cos(3x) = cos(2x+x) = cos(2x)cos(x) - sin(2x)sin(x) = cos^3(x) -sin^2(x)cos(x) - 2sin^2(x)cos(x) = cos^3(x) - 3sin^2(x)cos(x) =>

cos(3x) = cos^2(3x)

cos(3x)(cos(3x)-1) = 0

cos(3x) = 0 или cos(3x)=1

x = pi/6 +pik/2, k принадлежит Z или x = 2pi*n/3 , n принадлежит Z

Answer 2

Ah, that's a trigonometric equation! To solve it, you'll want to use some trigonometric identities. Let's break it down:

$4 \cos(x) \sin(30 + x) \cos(60 + x) = \cos^2(3x)$

First, let's express the angles in terms of known angles using trigonometric identities:

$\sin(30 + x) = \sin(30)\cos(x) + \cos(30)\sin(x)$ $\cos(60 + x) = \cos(60)\cos(x) - \sin(60)\sin(x)$

Now substitute these into the original equation and simplify:

$4 \cos(x) (\sin(30)\cos(x) + \cos(30)\sin(x)) (\cos(60)\cos(x) - \sin(60)\sin(x)) = \cos^2(3x)$

Expand and simplify both sides of the equation. Use trigonometric identities like $\sin(30) = \frac{1}{2}$, $\cos(30) = \frac{\sqrt{3}}{2}$, $\cos(60) = \frac{1}{2}$, and $\sin(60) = \frac{\sqrt{3}}{2}$.

After simplifying, you might end up with a quadratic equation in terms of trigonometric functions. From there, you can solve for $x$. It might get a bit messy, but with patience, you'll get there! If you need more help or want to share your progress, feel free!