Question

(корень 3\2)sinx+(1\2)cosx=sin5x

Answer 1

$\frac{\sqrt{3} }{2}Sinx+\frac{1}{2}Cosx=Sin5x\\\\Cos\frac{\pi }{6}Sinx+Sin\frac{\pi }{6}Cosx=Sin5x\\\\Sin(x+\frac{\pi }{6})-Sin5x=0\\\\2Sin\frac{x+\frac{\pi }{6}-5x }{2}Cos\frac{x+\frac{\pi }{6}+5x }{2} =0\\\\Sin(\frac{\pi }{12} -2x)Cos(3x+\frac{\pi }{12})=0\\\\1)Sin(\frac{\pi }{12}-2x)=0\\\\-2x+\frac{\pi }{12}=\pi n,n\in z\\\\-2x=-\frac{\pi }{12}+\pi n,n\in z\\\\x=\frac{\pi }{24} -\frac{\pi n }{2},n\in z$

$2)Cos(3x+\frac{\pi }{12})=0\\\\3x+\frac{\pi }{12}=\frac{\pi }{2} +\pi n,n\in z\\\\3x=\frac{\pi }{2}-\frac{\pi }{12}+\pi n,n\in z\\\\3x=\frac{5\pi }{12}+\pi n,n\in z\\\\x=\frac{5\pi }{36}+\frac{\pi n }{3},n\in z$

Answer 2

To solve the trigonometric equation $\sqrt{3/2} \sin(x) + \frac{1}{2} \cos(x) = \sin(5x)$, we'll need to apply trigonometric identities and algebraic manipulation. Let's go step by step:

Step 1: Express sin(5x) in terms of sin(x) and cos(x). We'll use the multiple-angle identity for sin(5x):

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

Step 2: Express cos(2x) and sin(2x) in terms of sin(x) and cos(x). Using the double-angle identities:

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

Step 3: Express cos(3x) in terms of sin(x) and cos(x). Using the triple-angle identity for cos(3x):

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

Step 4: Substitute the expressions from Step 2 and Step 3 into the expression for sin(5x) from Step 1.

$\sin(5x) = (\sin(3x))(2\sin(x)\cos(x)) + (4\cos^3(x) - 3\cos(x))(\cos(2x))$

Step 5: Simplify the expression.

$\sin(5x) = 2\sin(3x)\sin(x)\cos(x) + (4\cos^3(x) - 3\cos(x))(\cos^2(x) - \sin^2(x))$

Step 6: Use the Pythagorean identity to replace $\cos^2(x) - \sin^2(x)$.

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

Substitute this back into the equation:

$\sin(5x) = 2\sin(3x)\sin(x)\cos(x) + (4\cos^3(x) - 3\cos(x))(1 - 2\sin^2(x))$

Step 7: Express $\sin(3x)$ in terms of $\sin(x)$ and $\cos(x)$ using the triple-angle identity.

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

Substitute this back into the equation:

$\sin(5x) = 2(3\sin(x) - 4\sin^3(x))\sin(x)\cos(x) + (4\cos^3(x) - 3\cos(x))(1 - 2\sin^2(x))$

Step 8: Distribute and simplify:

$\sin(5x) = 6\sin^2(x)\cos(x) - 8\sin^4(x)\cos(x) + (4\cos^3(x) - 3\cos(x)) - 2(4\cos^3(x) - 3\cos(x))\sin^2(x)$

Step 9: Combine like terms:

$\sin(5x) = 6\sin^2(x)\cos(x) - 8\sin^4(x)\cos(x) + 4\cos^3(x) - 3\cos(x) - 8\cos^3(x)\sin^2(x) + 6\cos(x)\sin^2(x)$

Step 10: Bring all terms to one side of the equation:

$0 = 6\sin^2(x)\cos(x) - 8\sin^4(x)\cos(x) + 4\cos^3(x) - 3\cos(x) - 8\cos^3(x)\sin^2(x) + 6\cos(x)\sin^2(x) - \sin(5x)$