Question

Cos^2((pi/4) +3x) +5sin(6x) =-4

Answer 1

Ответ:

$x = -\frac{\pi}{12} + \frac{\pi n}{3}$

Пошаговое объяснение:

Применяем формулу двойного угла для косинуса. А затем формулу приведения.

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Ответ:  $x = -\frac{\pi}{12} + \frac{\pi n}{3}$

Формула двойного угла: $2cos^{2}(x) - 1 = cos (2x)$

Формула приведения: $cos(90^{0} + x) = - sin (x)$

Answer 2

The given equation is:

$\cos^2\left(\frac{\pi}{4} + 3x\right) + 5\sin(6x) = -4$

Let's solve this equation step by step:

1. Start by using the trigonometric identity $\cos^2(\theta) + \sin^2(\theta) = 1$ to simplify the left side of the equation:

$\cos^2\left(\frac{\pi}{4} + 3x\right) + 5\sin(6x) = 1 - \sin^2\left(\frac{\pi}{4} + 3x\right) + 5\sin(6x) = -4$

2. Rearrange the equation:

$-\sin^2\left(\frac{\pi}{4} + 3x\right) + 5\sin(6x) = -5$

3. Divide both sides of the equation by -1 to make the equation positive:

$\sin^2\left(\frac{\pi}{4} + 3x\right) - 5\sin(6x) = 5$

4. Use the trigonometric identity $\sin^2(\theta) = 1 - \cos^2(\theta)$ to express $\sin^2$ in terms of $\cos^2$:

$1 - \cos^2\left(\frac{\pi}{4} + 3x\right) - 5\sin(6x) = 5$

5. Now we have an equation in terms of $\cos$ and $\sin$:

$1 - \cos^2\left(\frac{\pi}{4} + 3x\right) - 5\sin(6x) = 5$

6. Use the trigonometric identity $\cos(\theta + \pi/4) = \frac{1}{\sqrt{2}} (\sin(\theta) + \cos(\theta))$ to express $\cos$ in terms of $\sin$:

$1 - \left(\frac{1}{\sqrt{2}} (\sin(3x) + \cos(3x))\right)^2 - 5\sin(6x) = 5$

7. Expand the squared term:

$1 - \frac{1}{2}(\sin^2(3x) + 2\sin(3x)\cos(3x) + \cos^2(3x)) - 5\sin(6x) = 5$

8. Simplify the trigonometric terms using the identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$:

$1 - \frac{1}{2}(\sin^2(3x) + \sin(6x)) - 5\sin(6x) = 5$

9. Combine the $\sin(6x)$ terms:

$1 - \frac{1}{2}\sin^2(3x) - \frac{3}{2}\sin(6x) = 5$

10. Rearrange the terms:

$-\frac{1}{2}\sin^2(3x) - \frac{3}{2}\sin(6x) = 4$

At this point, the equation is in a somewhat simplified form. However, it doesn't appear to have a straightforward algebraic solution for $x$ due to the combination of trigonometric functions and the squared sine term. Depending on the context, you might need to use numerical methods or graphing techniques to approximate the solutions for $x$.