Question

Cos(4x-30°)=1/3 sin(x+π/3)=1/2 tg(2x+15°)=кореньиз2/2

Answer 1

Ответ:

1.

$\cos(4x - 30°) = \frac{1}{3} \\ \\ 4x 1- 30 °= arccos( \frac{1}{3} ) + 2\pi \: n \\ 4x1 = 30 °+ arccos( \frac{1}{3} ) + 2\pi \: n \\ x1 = 7.5° + \frac{1}{4}arccos( \frac{1}{3} ) + \frac{\pi \: n}{2} \\ \\ 4x2 - 30 °= - arccos( \frac{1}{3} ) + 2\pi \: n \\ x2 =7 .5 °- \frac{1}{4} arccos( \frac{1}{3} ) + \frac{\pi \: n}{2}$

n принадлежит Z.

2.

$\sin(x + \frac{\pi}{3} ) = \frac{1}{2} \\ \\ x1 + \frac{\pi}{3} = \frac{\pi}{6} + 2\pi\ \: n \\ x1 = - \frac{\pi}{6} + 2\pi \: n \\ \\ x2 + \frac{\pi}{3} = \frac{5\pi}{6} + 2\pi \: n \\ x2 = \frac{\pi}{2} + 2\pi \: n$

n принадлежит Z.

3.

$tg(2x + 15°) = \frac{ \sqrt{2} }{2} \\ 2x + 15° = \frac{\pi}{4} + \pi \: n \\ 2x = - 15° + 45 °+ \pi \: n \\ 2x = 30° + \pi \: n \\ 2x = \frac{\pi}{6} + \pi \: n \\ x = \frac{\pi}{12} + \frac{\pi \: n}{2}$

n принадлежит Z.

Answer 2

To solve these trigonometric equations, we'll use trigonometric identities and properties. Let's solve each equation one by one:

1. $\cos(4x - 30°) = \frac{1}{3}$

To solve this equation, first, find the inverse cosine of $\frac{1}{3}$:

$4x - 30° = \cos^{-1}\left(\frac{1}{3}\right)$ $4x - 30° = \arccos\left(\frac{1}{3}\right)$ $4x - 30° = 70.53°$ (approximately)

Solve for $x$:

$4x = 100.53°$ $x \approx 25.13°$ (approximately)

2. $\sin(x + \frac{\pi}{3}) = \frac{1}{2}$

To solve this equation, first, find the inverse sine of $\frac{1}{2}$:

$x + \frac{\pi}{3} = \sin^{-1}\left(\frac{1}{2}\right)$ $x + \frac{\pi}{3} = \arcsin\left(\frac{1}{2}\right)$ $x + \frac{\pi}{3} = 30°$

Solve for $x$:

$x = 30° - \frac{\pi}{3}$ $x = 30° - 60°$ $x = -30°$

3. $\tan(2x + 15°) = \frac{\sqrt{2}}{2}$

To solve this equation, first, find the inverse tangent of $\frac{\sqrt{2}}{2}$:

$2x + 15° = \tan^{-1}\left(\frac{\sqrt{2}}{2}\right)$ $2x + 15° = \arctan\left(\frac{\sqrt{2}}{2}\right)$ $2x + 15° = 45°$

Solve for $x$:

$2x = 30°$ $x = 15°$

So the solutions for the given equations are:

1. $x \approx 25.13°$ (approximately)
2. $x = -30°$
3. $x = 15°$