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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.

Solve the following equations: a) cos(2x - π) = cos(π/3), b) 4cos^2(x) - 3 = 0

Learn how to solve trigonometric equations like cos(2x - π) = cos(π/3) and quadratic equations involving trigonometric functions, such as 4cos^2(x) - 3 = 0. This solution involves trigonometric identities and general solutions for cosine. Suitable for Grades 10-12.

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