Find the smallest positive value of `theta` satisfying the equation sec 6 `theta` = cosec 3 `theta`

Answer

To solve the equation \( \sec 6\theta = \csc 3\theta \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
\[ \sec 6\theta = \frac{1}{\cos 6\theta} \quad \text{and} \quad \csc 3\theta = \frac{1}{\sin 3\theta} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\cos 6\theta} = \frac{1}{\sin 3\theta} \]

### Step 2: Cross-multiply
By cross-multiplying, we get:
\[ \sin 3\theta = \cos 6\theta \]

### Step 3: Use the co-function identity
Using the identity \( \sin x = \cos(90^\circ - x) \), we can express \( \sin 3\theta \) as:
\[ \sin 3\theta = \cos(90^\circ - 3\theta) \]
Thus, we can set the two expressions equal:
\[ \cos(90^\circ - 3\theta) = \cos 6\theta \]

### Step 4: Set up the equation
From the property of cosine, we know that:
\[ 90^\circ - 3\theta = 6\theta + n \cdot 360^\circ \quad \text{or} \quad 90^\circ - 3\theta = -6\theta + n \cdot 360^\circ \]
where \( n \) is any integer.

### Step 5: Solve the first equation
For the first equation:
\[ 90^\circ - 3\theta = 6\theta \]
Rearranging gives:
\[ 90^\circ = 9\theta \quad \Rightarrow \quad \theta = \frac{90^\circ}{9} = 10^\circ \]

### Step 6: Solve the second equation
For the second equation:
\[ 90^\circ - 3\theta = -6\theta \]
Rearranging gives:
\[ 90^\circ = -3\theta + 6\theta \quad \Rightarrow \quad 90^\circ = 3\theta \quad \Rightarrow \quad \theta = \frac{90^\circ}{3} = 30^\circ \]

### Step 7: Determine the smallest positive value
The two values we found are \( \theta = 10^\circ \) and \( \theta = 30^\circ \). The smallest positive value is:
\[ \theta = 10^\circ \]

### Final Answer:
The smallest positive value of \( \theta \) satisfying the equation \( \sec 6\theta = \csc 3\theta \) is \( \theta = 10^\circ \).

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