The problem is to solve the equation $sin(5x^2) = cos(2x+6)$. The angle is in degrees.

1. Problem Description

The problem is to solve the equation $sin(5x^2) = cos(2x+6)$. The angle is in degrees.

2. Solution Steps

We know that $sin(\theta) = cos(90 - \theta)$. Using this property, we can rewrite the given equation.

$sin(5x^2) = cos(2x+6)$

$cos(90-5x^2) = cos(2x+6)$

Since the cosine values are equal, we can equate the angles:

$90 - 5x^2 = 2x + 6$

Rearranging the terms, we get a quadratic equation:

$5x^2 + 2x - 84 = 0$

We can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a=5$, $b=2$, and $c=-84$.

$x = \frac{-2 \pm \sqrt{2^2 - 4(5)(-84)}}{2(5)}$

$x = \frac{-2 \pm \sqrt{4 + 1680}}{10}$

$x = \frac{-2 \pm \sqrt{1684}}{10}$

$x = \frac{-2 \pm 2\sqrt{421}}{10}$

$x = \frac{-1 \pm \sqrt{421}}{5}$

So the two possible solutions are:

$x = \frac{-1 + \sqrt{421}}{5}$ and $x = \frac{-1 - \sqrt{421}}{5}$

Approximating $\sqrt{421} \approx 20.518$

$x \approx \frac{-1 + 20.518}{5} \approx \frac{19.518}{5} \approx 3.9036$

$x \approx \frac{-1 - 20.518}{5} \approx \frac{-21.518}{5} \approx -4.3036$

3. Final Answer

The solutions are $x = \frac{-1 + \sqrt{421}}{5}$ and $x = \frac{-1 - \sqrt{421}}{5}$.

$x \approx 3.9036$ and $x \approx -4.3036$.