Question

Tg(3x+π/4)+ctg(3x+π/4)=2

Answer 1

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Answer 2

To solve the trigonometric equation $\tan(3x + \frac{\pi}{4}) + \cot(3x + \frac{\pi}{4}) = 2$, you can follow these steps:

1. First, recognize that $\cot(3x + \frac{\pi}{4})$ is the reciprocal of $\tan(3x + \frac{\pi}{4})$, so you can rewrite the equation as:

$\tan(3x + \frac{\pi}{4}) + \frac{1}{\tan(3x + \frac{\pi}{4})} = 2$

2. To simplify the equation further, multiply both sides by $\tan(3x + \frac{\pi}{4})$ to clear the fractions:

$\tan^2(3x + \frac{\pi}{4}) + 1 = 2\tan(3x + \frac{\pi}{4})$

3. Next, subtract $2\tan(3x + \frac{\pi}{4})$ from both sides:

$\tan^2(3x + \frac{\pi}{4}) - 2\tan(3x + \frac{\pi}{4}) + 1 = 0$

4. Now, you have a quadratic equation in terms of $\tan(3x + \frac{\pi}{4})$. You can substitute $t = \tan(3x + \frac{\pi}{4})$ to make it easier to solve:

$t^2 - 2t + 1 = 0$

5. This is a simple quadratic equation that factors as:

$(t - 1)^2 = 0$

6. Take the square root of both sides:

$t - 1 = 0$

7. Add 1 to both sides to solve for $t$:

$t = 1$

8. Now, you need to find the values of $x$. Recall that $t = \tan(3x + \frac{\pi}{4})$, so:

$\tan(3x + \frac{\pi}{4}) = 1$

9. To find the values of $x$, you can use inverse trigonometric functions. Take the arctan (inverse tangent) of both sides:

$3x + \frac{\pi}{4} = \arctan(1) = \frac{\pi}{4}$

10. Solve for $x$:

$3x = 0$

$x = 0$

So, the solution to the equation $\tan(3x + \frac{\pi}{4}) + \cot(3x + \frac{\pi}{4}) = 2$ is $x = 0$.