Question

Cos(3x) = 2sin((3π / 2) + x)

Answer 1

Ответ:

x= 60⁰+k×180⁰

90⁰+k×180⁰ k э z

120⁰+k×180⁰

Answer 2

Ответ:     π/2 + πn ;  ± π/3 + πn ,  nЄ Z .

Пошаговое объяснение:

cos(3x) = 2sin((3π / 2) + x) ;

cos(3x) = - 2cosx ;

cos3x + cosx = - cosx ;

2cos2xcosx + cosx = 0 ;

cosx( 2cos2x + 1 ) = 0 ;

cosx = 0                             2cos2x + 1 = 0 ;

x = π/2 + πn , nЄ Z ;                    cos2x = - 1/2 ;

                                                  2x = ± ( π - π/3 ) + 2πn , nЄ Z ;

                                                  2x = ± 2π/3 + 2πn ;

                                                    x = ± π/3 + πn ,  nЄ Z .                

Answer 3

This looks like a trigonometric equation! To solve it, we can use some trigonometric identities. Let's see if we can simplify it.

The given equation is:

$\cos(3x) = 2\sin\left(\frac{3\pi}{2} + x\right)$

First, we know that $\cos(3x)$ can be expressed using the double-angle formula as $4\cos^3(x) - 3\cos(x)$. So, the equation becomes:

$4\cos^3(x) - 3\cos(x) = 2\sin\left(\frac{3\pi}{2} + x\right)$

Now, let's express $\sin\left(\frac{3\pi}{2} + x\right)$ using the angle sum identity:

$4\cos^3(x) - 3\cos(x) = 2\cos(x)\sin\left(\frac{3\pi}{2}\right) + 2\sin(x)\cos\left(\frac{3\pi}{2}\right)$

Since $\sin\left(\frac{3\pi}{2}\right) = -1$ and $\cos\left(\frac{3\pi}{2}\right) = 0$, the equation simplifies to:

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

Now, you can solve this cubic equation for $\cos(x)$. Once you find the values of $\cos(x)$, you can use them to determine $x$ by taking the inverse cosine.

Remember that the solutions might involve multiple steps, and it's always a good idea to double-check your work!