The image contains two trigonometric equations in Finnish:

Task: "Ratkaise yhtälö."
Translation: "Solve the equation."

The given equations are:

a) $\cos(2x - \pi) = \cos\left(\frac{\pi}{3}\right)$
b) $4 \cos^2 x - 3 = 0$

Let's solve these equations step by step:

a) $\cos(2x - \pi) = \cos\left(\frac{\pi}{3}\right)$

First, recall the identity for the cosine function: $\cos(\theta) = \cos(2k\pi \pm \theta)$ The cosine of an angle shifted by $\pi$ satisfies the following property: $\cos(2x - \pi) = -\cos(2x)$

So, we can rewrite the equation as: $-\cos(2x) = \cos\left(\frac{\pi}{3}\right)$

From trigonometry, $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$, so the equation becomes: $-\cos(2x) = \frac{1}{2}$ $\cos(2x) = -\frac{1}{2}$

Now, solve for $2x$: $\cos(2x) = -\frac{1}{2}$

The general solution for $\cos(2x) = -\frac{1}{2}$ is: $2x = 2k\pi \pm \frac{2\pi}{3}$

Thus, divide both sides by 2 to find $x$: $x = k\pi \pm \frac{\pi}{3}$

This gives the general solution for $x$: $x = k\pi \pm \frac{\pi}{3}, \quad k \in \mathbb{Z}$

b) $4 \cos^2 x - 3 = 0$

Rearrange the equation: $4 \cos^2 x = 3$

Divide by 4: $\cos^2 x = \frac{3}{4}$

Take the square root of both sides: $\cos x = \pm \frac{\sqrt{3}}{2}$

Now, solve for $x$. The solutions for $\cos x = \frac{\sqrt{3}}{2}$ are: $x = 2k\pi \pm \frac{\pi}{6}, \quad k \in \mathbb{Z}$

And for $\cos x = -\frac{\sqrt{3}}{2}$: $x = 2k\pi \pm \frac{5\pi}{6}, \quad k \in \mathbb{Z}$

Summary of Solutions:

• a) $x = k\pi \pm \frac{\pi}{3}, \quad k \in \mathbb{Z}$
• b) $x = 2k\pi \pm \frac{\pi}{6}$ and $x = 2k\pi \pm \frac{5\pi}{6}, \quad k \in \mathbb{Z}$

Would you like further clarification or more details?

Here are 5 related questions for further exploration:

1. How do we derive the general solution for cosine equations?
2. What is the significance of the trigonometric identities used here?
3. Can cosine equations with different angles, like $\cos(2x)$, have the same periodicity as $\cos(x)$?
4. How do sine and cosine equations differ in terms of their general solutions?
5. What are some real-world applications of solving trigonometric equations?

Tip: When solving trigonometric equations, always check for all possible solutions within the given interval, as cosine and sine functions are periodic.