Question

Tg x =√3 ............​

Answer 1

x=arctg3+pk

$x = \frac{\pi}{3} + \pi \: k \:$

kпринадлежит множеству целых чисел

Answer 2

To solve the equation $\tan(x) = \sqrt{3}$, we need to find the value of $x$ that satisfies the equation. Here's the step-by-step solution:

Step 1: Recall the relationship between tangent and sine/cosine functions: $\tan(x) = \frac{\sin(x)}{\cos(x)}$

Step 2: Since we know that $\tan(x) = \sqrt{3}$, we can rewrite the equation as: $\frac{\sin(x)}{\cos(x)} = \sqrt{3}$

Step 3: To get rid of the fraction, we can multiply both sides by $\cos(x)$: $\sin(x) = \sqrt{3} \cdot \cos(x)$

Step 4: Now, remember the Pythagorean identity for trigonometric functions: $\sin^2(x) + \cos^2(x) = 1$

Step 5: We can use this identity to express $\cos(x)$ in terms of $\sin(x)$: $\cos^2(x) = 1 - \sin^2(x)$

Step 6: Substitute this expression for $\cos^2(x)$ into the equation from Step 3: $\sin(x) = \sqrt{3} \cdot \sqrt{1 - \sin^2(x)}$

Step 7: Square both sides to eliminate the square roots: $\sin^2(x) = 3 \cdot (1 - \sin^2(x))$

Step 8: Expand the right-hand side: $\sin^2(x) = 3 - 3\sin^2(x)$

Step 9: Move all terms containing $\sin^2(x)$ to one side of the equation: $4\sin^2(x) = 3$

Step 10: Divide both sides by 4 to solve for $\sin^2(x)$: $\sin^2(x) = \frac{3}{4}$

Step 11: Now, take the square root of both sides to find $\sin(x)$: $\sin(x) = \pm \frac{\sqrt{3}}{2}$

Step 12: Since $\sin(x)$ is positive in both the first and second quadrants, we can ignore the negative solution: $\sin(x) = \frac{\sqrt{3}}{2}$

Step 13: Finally, find the value of $x$ using the inverse sine (arcsin) function: $x = \arcsin\left(\frac{\sqrt{3}}{2}\right)$

The principal value of $x$ that satisfies $\tan(x) = \sqrt{3}$ is $x = \frac{\pi}{3}$ (or $x = 60^\circ$ in degrees). However, it's important to note that the trigonometric functions have periodic behavior, and there are infinitely many solutions for $x$ due to the periodicity of the tangent function. The general solution is given by $x = \frac{\pi}{3} + n\pi$, where $n$ is an integer.