https://file.aiquickdraw.com/mathbot/file/17266832502879b6qc34x.jpg

Let's solve the given trigonometric equations step by step.

### (i) \( 2 \sin^2 2x = \sqrt{2} \sin 2x \)

1. **Rearrange the equation**: 
   \[
   2 \sin^2 2x - \sqrt{2} \sin 2x = 0
   \]
   Factor the equation:
   \[
   \sin 2x (2 \sin 2x - \sqrt{2}) = 0
   \]
2. **Solve for \(\sin 2x = 0\)**:
   \[
   \sin 2x = 0
   \]
   The general solutions for \(\sin \theta = 0\) are \( \theta = n\pi \), where \(n\) is an integer. For \(0^\circ \leq x \leq 360^\circ\):
   \[
   2x = 0^\circ, 180^\circ, 360^\circ \quad \Rightarrow \quad x = 0^\circ, 90^\circ, 180^\circ
   \]

3. **Solve for \(2 \sin 2x = \sqrt{2}\)**:
   \[
   \sin 2x = \frac{\sqrt{2}}{2}
   \]
   This occurs when \( 2x = 45^\circ, 135^\circ\) in one full cycle. Thus:
   \[
   x = 22.5^\circ, 67.5^\circ
   \]

   Final solutions for (i): \( x = 0^\circ, 90^\circ, 180^\circ, 22.5^\circ, 67.5^\circ \).

---

### (ii) \( 3 \tan x = \frac{1}{\cos x} \)

1. **Rewrite using the identity \( \frac{1}{\cos x} = \sec x \)**:
   \[
   3 \tan x = \sec x
   \]
   Divide both sides by \(\cos x\) (since \(\sec x = \frac{1}{\cos x}\)):
   \[
   3 \sin x = 1
   \]
   Solving for \(\sin x\):
   \[
   \sin x = \frac{1}{3}
   \]
2. **Find general solutions for \(\sin x = \frac{1}{3}\)**:
   This occurs when:
   \[
   x = \arcsin \left( \frac{1}{3} \right) \approx 19.47^\circ
   \]
   The general solutions for sine are:
   \[
   x = 19.47^\circ, 180^\circ - 19.47^\circ = 160.53^\circ
   \]
   
   Final solutions for (ii): \( x = 19.47^\circ, 160.53^\circ \).

---

### (iii) \( 6 \sin^2 x - 7 \sin x = -2 \)

1. **Rearrange into a quadratic equation**:
   \[
   6 \sin^2 x - 7 \sin x + 2 = 0
   \]
   Factor the quadratic:
   \[
   (3 \sin x - 2)(2 \sin x - 1) = 0
   \]
2. **Solve each factor**:
   - \( 3 \sin x - 2 = 0 \quad \Rightarrow \quad \sin x = \frac{2}{3} \)
   - \( 2 \sin x - 1 = 0 \quad \Rightarrow \quad \sin x = \frac{1}{2} \)

3. **Find general solutions for \(\sin x = \frac{2}{3}\)**:
   \[
   x = \arcsin \left( \frac{2}{3} \right) \approx 41.81^\circ
   \]
   And the second solution:
   \[
   x = 180^\circ - 41.81^\circ = 138.19^\circ
   \]

4. **Find general solutions for \(\sin x = \frac{1}{2}\)**:
   \[
   x = 30^\circ, 150^\circ
   \]

   Final solutions for (iii): \( x = 41.81^\circ, 138.19^\circ, 30^\circ, 150^\circ \).

---

### (iv) \( 3 \tan 2x = \frac{1}{\tan 2x} \)

1. **Multiply both sides by \(\tan 2x\)**:
   \[
   3 \tan^2 2x = 1
   \]
   Solve for \(\tan^2 2x\):
   \[
   \tan^2 2x = \frac{1}{3}
   \quad \Rightarrow \quad \tan 2x = \pm \frac{1}{\sqrt{3}}
   \]
2. **Solve for \(2x\)**:
   - For \( \tan 2x = \frac{1}{\sqrt{3}} \), we have \( 2x = 30^\circ, 210^\circ \).
   - For \( \tan 2x = -\frac{1}{\sqrt{3}} \), we have \( 2x = 150^\circ, 330^\circ \).
   
   Divide by 2 to find \(x\):
   \[
   x = 15^\circ, 75^\circ, 105^\circ, 165^\circ
   \]

   Final solutions for (iv): \( x = 15^\circ, 75^\circ, 105^\circ, 165^\circ \).

---

Let me know if you'd like further details on any of these steps or have additional questions!

### 5 Related Questions:
1. How do you solve quadratic trigonometric equations?
2. What are the general solutions for \(\sin \theta = 0\)?
3. How do you apply the factorization method to trigonometric equations?
4. What is the relationship between \(\sec x\) and \(\cos x\)?
5. How do you solve equations involving \(\tan^2 x\)?

### Tip:
For trigonometric equations, always check if a factorization technique or an identity can simplify the equation first before solving for angles.

Solve the following equations, giving all solutions such that 0° ≤ x ≤ 360°: (i) 2sin²2x = √2sin2x (ii) 3tanx = 1/cosx (iii) 6sin²x - 7sinx = -2 (iv) 3tan2x = 1/tan2x

Learn how to solve various trigonometric equations like 2sin²2x = √2sin2x and 3tan2x = 1/tan2x. This problem covers key trigonometric concepts, including identities, factorization, and general solutions for sine and tangent functions. Suitable for Grades 10-12 students.

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