If `sec(7theta+28^(@))=cosec(30^(@)-3theta)` then the value of `theta` is

Options

A. `8^(@)`
B. `5^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To solve the equation \( \sec(7\theta + 28^\circ) = \csc(30^\circ - 3\theta) \), we can follow these steps:

### Step 1: Rewrite the cosecant function
We know that \( \csc x = \sec(90^\circ - x) \). Therefore, we can rewrite the equation as:
\[ \sec(7\theta + 28^\circ) = \sec(90^\circ - (30^\circ - 3\theta)) \]

### Step 2: Simplify the right side
Now, simplify the right side:
\[ 90^\circ - (30^\circ - 3\theta) = 90^\circ - 30^\circ + 3\theta = 60^\circ + 3\theta \]
Thus, we have:
\[ \sec(7\theta + 28^\circ) = \sec(60^\circ + 3\theta) \]

### Step 3: Set the angles equal
Since the secant function is equal when the angles are equal (or differ by multiples of \(180^\circ\)), we can write:
\[ 7\theta + 28^\circ = 60^\circ + 3\theta \]

### Step 4: Rearrange the equation
Now, rearranging the equation gives:
\[ 7\theta - 3\theta = 60^\circ - 28^\circ \]
This simplifies to:
\[ 4\theta = 32^\circ \]

### Step 5: Solve for \( \theta \)
Now, divide both sides by 4:
\[ \theta = \frac{32^\circ}{4} = 8^\circ \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 8^\circ \]

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