Question

1) 2синус X/2 косинус 5к-косинус х = 0 2) tg(30°-Х)=√33) косинус(π/4-2х)= -1 4) tg(4х-π/6)=√3/3

Answer 1

Под 3 только знаю сорян(

Answer 2

To solve the given trigonometric equations, we'll find the values of the variable 'x' that satisfy each equation.

1. $\frac{2\sin(\frac{x}{2}) \cos(5k) - \cos(x)}{2} = 0$

To solve this equation, we'll first try to simplify it:

Step 1: Use the identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ $\frac{\sin(x)\cos(5k) - \cos(x)}{2} = 0$

Step 2: Find a common denominator and combine the terms: $\frac{\sin(x)\cos(5k) - 2\cos(x)}{2} = 0$

Step 3: Factor out common terms from the numerator: $\frac{\cos(5k)\sin(x) - 2\cos(x)}{2} = 0$

Now, to find the values of 'x', we need to set the numerator equal to zero:

$\cos(5k)\sin(x) - 2\cos(x) = 0$

2. $\tan(30° - x) = \sqrt{3}$

Step 1: Use the identity $\tan(30°) = \frac{\sqrt{3}}{3}$ to rewrite the equation: $\tan(30°)\cot(x) - \cot(x) = \sqrt{3}$

Step 2: Factor out $\cot(x)$: $\cot(x) (\tan(30°) - 1) = \sqrt{3}$

Step 3: Use the value of $\tan(30°)$: $\frac{1}{\sqrt{3}} \cot(x) = \sqrt{3}$

Step 4: Solve for $\cot(x)$: $\cot(x) = \sqrt{3} \cdot \sqrt{3} = 3$

3. $\cos(\frac{\pi}{4} - 2x) = -1$

Step 1: Use the identity $\cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ to rewrite the equation: $\cos(\frac{\pi}{4})\sin(2x) + \sin(\frac{\pi}{4})\cos(2x) = -1$

Step 2: Use the double angle formulas $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ and $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$: $\frac{1}{\sqrt{2}}(2\sin(x)\cos(x) + \cos^2(x) - \sin^2(x)) = -1$

Step 3: Use the identity $\cos^2(x) + \sin^2(x) = 1$: $\frac{1}{\sqrt{2}}(2\sin(x)\cos(x) + 1 - \sin^2(x)) = -1$