Consider the following statements :
1. `3theta` is an acute angle such that sin `3theta=cos2theta` , then the measurement of `theta` in radian equals to `(pi)/(10)` .
2. One radian is the angle subtended at the centre of a circle by an arc of the same circle by an arc of the same circle whose length is equal to the diameter of that circle.
Which of the above statements is /are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 3
D. Neither 1 nor 3

Explanation

To determine the correctness of the given statements, we will analyze each statement step by step.

### Step 1: Analyze the first statement
The first statement is:
**"3θ is an acute angle such that sin(3θ) = cos(2θ), then the measurement of θ in radian equals to (π)/(10)."**

We know that:
\[ \sin(3θ) = \cos(2θ) \]

Using the co-function identity, we can rewrite this as:
\[ \sin(3θ) = \sin\left(\frac{\pi}{2} - 2θ\right) \]

### Step 2: Set up the equation
Since the sine function is equal, we can set up the equation:
\[ 3θ = \frac{\pi}{2} - 2θ \]

### Step 3: Solve for θ
Now, we will solve for θ:
\[ 3θ + 2θ = \frac{\pi}{2} \]
\[ 5θ = \frac{\pi}{2} \]
\[ θ = \frac{\pi}{10} \]

### Step 4: Conclusion for the first statement
Thus, the first statement is correct as we found that θ in radians equals to \(\frac{\pi}{10}\).

### Step 5: Analyze the second statement
The second statement is:
**"One radian is the angle subtended at the center of a circle by an arc of the same circle whose length is equal to the diameter of that circle."**

### Step 6: Recall the definition of one radian
The correct definition of one radian is:
One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle, not the diameter.

### Step 7: Conclusion for the second statement
Since the statement incorrectly states that the arc length is equal to the diameter instead of the radius, the second statement is false.

### Final Conclusion
- The first statement is correct.
- The second statement is incorrect.

Thus, the answer is:
**Option A: "One only."**

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