Question

3cos(3π/2+x)-5cosx=0​

Answer 1

Используя формулы приведения cos(3π/2 + x) = sin x

3sin x - 5cos x = 0

3sin x = 5cos x |:cos x ≠ 0

3tg x = 5

tg x = 5/3

x = arctg(5/3) + πn, n ∈ Z

Answer 2

To solve the trigonometric equation $3\cos\left(\frac{3\pi}{2} + x\right) - 5\cos(x) = 0$, we can use trigonometric identities. Let's break down the problem step by step:

1. Use Angle Sum Identity: Recall the angle sum identity for cosine: $\cos(A + B) = \cos A \cos B - \sin A \sin B$.

2. Apply the Identity to the Equation: Apply the angle sum identity to $3\cos\left(\frac{3\pi}{2} + x\right) - 5\cos(x) = 0$.

   $3\left(\cos\frac{3\pi}{2}\cos x - \sin\frac{3\pi}{2}\sin x\right) - 5\cos x = 0$

   Since $\cos\frac{3\pi}{2} = 0$ and $\sin\frac{3\pi}{2} = -1$, this simplifies to:

   $-3\sin x - 5\cos x = 0$

3. Combine Terms and Rearrange: Rearrange the equation by combining like terms:

   $-5\cos x - 3\sin x = 0$

4. Use Pythagorean Identity: Recall the Pythagorean identity for sine and cosine: $\sin^2 x + \cos^2 x = 1$.

   Divide the entire equation by $\sqrt{(-5)^2 + (-3)^2} = \sqrt{34}$ to normalize the coefficients:

   $-\frac{5}{\sqrt{34}}\cos x - \frac{3}{\sqrt{34}}\sin x = 0$

5. Express in Standard Form: Express the equation in standard form by multiplying through by $-\sqrt{34}$:

   $-5\cos x - 3\sin x = 0$

6. Use Trigonometric Ratio Identity: The equation can now be written as $\tan x = -\frac{5}{3}$.

7. Find the Angle $x$: Find the angle $x$ by taking the arctangent of $-\frac{5}{3}$:

   $x = \arctan\left(-\frac{5}{3}\right)$

So, the general solution for the given trigonometric equation is x = \arctan\left(-\frac{5}{3} + $n\pi$, where $n$ is an integer. Keep in mind that the exact value of $\arctan\left(-\frac{5}{3}\right)$ may need to be expressed in terms of radicals or in decimal form.