If `sin(20^(@)+theta)=cos30^(@)` then find the value of `theta`.

Answer

To solve the equation \( \sin(20^\circ + \theta) = \cos(30^\circ) \), we can follow these steps:

### Step 1: Rewrite \( \cos(30^\circ) \)
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
However, we will use the complementary angle identity for sine:
\[ \cos(30^\circ) = \sin(90^\circ - 30^\circ) = \sin(60^\circ) \]

### Step 2: Set the equation
Now we can rewrite the original equation:
\[ \sin(20^\circ + \theta) = \sin(60^\circ) \]

### Step 3: Use the property of sine
Since \( \sin A = \sin B \) implies that \( A = B + n \cdot 360^\circ \) or \( A = 180^\circ - B + n \cdot 360^\circ \) (where \( n \) is any integer), we can set up the following equations:
1. \( 20^\circ + \theta = 60^\circ + n \cdot 360^\circ \)
2. \( 20^\circ + \theta = 180^\circ - 60^\circ + n \cdot 360^\circ \)

### Step 4: Solve the first equation
From the first equation:
\[ 20^\circ + \theta = 60^\circ \]
Subtract \( 20^\circ \) from both sides:
\[ \theta = 60^\circ - 20^\circ = 40^\circ \]

### Step 5: Solve the second equation
From the second equation:
\[ 20^\circ + \theta = 120^\circ \]
Subtract \( 20^\circ \) from both sides:
\[ \theta = 120^\circ - 20^\circ = 100^\circ \]

### Step 6: Determine the valid solution
Since \( \theta \) is typically considered in the range of \( 0^\circ \) to \( 90^\circ \) for this context, we take:
\[ \theta = 40^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 40^\circ \]

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