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When solving trigonometric equations, the first step is often to isolate the trigonometric function. It means you need to get the trigonometric part (like \(\text{cos}(2x)\)) by itself on one side of the equation.

For example, let's look at the equation:\[2 \text{cos}(2x) = -\sqrt{2}\]
To isolate \(\text{cos}(2x)\), we divide both sides by 2: \[ \text{cos}(2x) = -\frac{\sqrt{2}}{2} \]
The equation is now simpler and ready for the next steps.

After isolating the trigonometric function, we need to identify the reference angles. A reference angle is a standard angle that shares the same trigonometric properties within a given interval.

In this case, we need to find the angles whose cosine equals \(-\frac{\sqrt{2}}{2}\). In the interval \([0, 2\pi)\), these angles are \( \frac{3\pi}{4} \) and \(\frac{5\pi}{4}\). Hence, \(\text{cos}(\theta) = -\frac{\sqrt{2}}{2}\) when \(\theta \) is either of these angles.

These reference angles are the key to finding the complete solution since they help us to express the general solution for the original equation.

The cosine function is pivotal in trigonometry. It represents the relationship between the angle and the adjacent side over the hypotenuse in a right triangle.

When you solve equations involving the cosine function, you'll often find yourself working within specific intervals, typically \([0, 2\pi)\). For instance, once we isolated \(\text{cos}(2x)\) and identified our reference angles, we set \(\text{cos}(2x)\) to those angles: \[2x = \frac{3\pi}{4} + 2k\pi \] and \[2x = \frac{5\pi}{4} + 2k\pi\]

Next, we solve for \(x\) by dividing both sides of each equation by 2, yielding: \[ x = \frac{3\pi}{8} + k\pi \] and \[ x = \frac{5\pi}{8} + k\pi\] where \(k\) is any integer.

Understanding the properties and behavior of the cosine function helps you see why we set up our solutions this way and ensures you can find all possible solutions.