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Let’s solve each trigonometric equation within the given interval \( 0 \leq x \leq 2\pi \).

### Problem (a)
\[
\cos(2x) + 3 + 4\cos(x) = 0
\]

#### Solution:
1. Use the double-angle identity: \(\cos(2x) = 2\cos^2(x) - 1\).
   Substitute this into the equation:
   \[
   2\cos^2(x) - 1 + 3 + 4\cos(x) = 0
   \]
2. Simplify the equation:
   \[
   2\cos^2(x) + 4\cos(x) + 2 = 0
   \]
3. Divide through by 2:
   \[
   \cos^2(x) + 2\cos(x) + 1 = 0
   \]
4. Factorize:
   \[
   (\cos(x) + 1)^2 = 0
   \]
5. Solve for \(\cos(x)\):
   \[
   \cos(x) = -1
   \]
6. The solution in the interval \(0 \leq x \leq 2\pi\) is:
   \[
   x = \pi
   \]

### Problem (b)
\[
2\cos(2x) + 1 + \sin(x) = 0
\]

#### Solution:
1. Use the double-angle identity: \(\cos(2x) = 2\cos^2(x) - 1\).
   Substitute this into the equation:
   \[
   2(2\cos^2(x) - 1) + 1 + \sin(x) = 0
   \]
2. Expand and simplify:
   \[
   4\cos^2(x) - 2 + 1 + \sin(x) = 0
   \]
   \[
   4\cos^2(x) + \sin(x) - 1 = 0
   \]
3. Substitute \(\sin(x) = \sqrt{1 - \cos^2(x)}\) and solve for values of \(x\) that satisfy this equation in the interval.

### Problem (c)
\[
\tan(2x) + 5\tan(x) = 0
\]

#### Solution:
1. Use the double-angle identity: \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\).
2. Substitute and solve for values of \(x\) in the interval.

Would you like more detailed steps for any of these problems or have any specific questions?

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Here are some related questions to deepen understanding:
1. How can we factorize equations involving trigonometric identities?
2. What are the common trigonometric identities used for solving equations?
3. How do we determine which identity to use based on the given equation?
4. What are the specific intervals where certain trigonometric functions are positive or negative?
5. How can we solve trigonometric equations with both sine and cosine terms?

**Tip:** When solving trigonometric equations, it’s helpful to first try rewriting the equation using identities, simplifying wherever possible.

Solve these equations for 0 ≤ x ≤ 2π:

a) cos(2x) + 3 + 4cos(x) = 0

b) 2cos(2x) + 1 + sin(x) = 0

c) tan(2x) + 5tan(x) = 0

Step-by-step solution to solve trigonometric equations involving cosine, sine, and tangent in the interval 0 ≤ x ≤ 2π. Includes use of trigonometric identities and double-angle formulas for cosine and tangent. Suitable for students in Grades 11-12.

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