The graph of `y=sin2x` for x between `10^(@)` and `350^(@)` crosses the x-axis

Options

A. zero times
B. one time
C. two times
D. three times

Explanation

To determine how many times the graph of \( y = \sin(2x) \) crosses the x-axis for \( x \) between \( 10^\circ \) and \( 350^\circ \), we need to find the values of \( x \) where \( y = 0 \). This occurs when \( \sin(2x) = 0 \).

### Step-by-Step Solution:

1. **Set the equation to zero**:
\[ \sin(2x) = 0 \]

2. **Find the general solutions**:
The sine function is zero at integer multiples of \( \pi \):
\[ 2x = n\pi \quad \text{where } n \text{ is an integer} \]
Thus, we can express \( x \) as:
\[ x = \frac{n\pi}{2} \]

3. **Convert \( n \) to degrees**:
Since we are working in degrees, we convert \( \pi \) to degrees:
\[ \pi \text{ radians} = 180^\circ \]
Therefore, the equation becomes:
\[ x = \frac{n \cdot 180^\circ}{2} = 90n \]

4. **Determine the values of \( n \)**:
We need to find the values of \( n \) such that \( x \) lies between \( 10^\circ \) and \( 350^\circ \):
\[ 10^\circ < 90n < 350^\circ \]

5. **Solve the inequalities**:
- For the lower bound:
\[ 90n > 10 \implies n > \frac{10}{90} \implies n > \frac{1}{9} \implies n \geq 1 \quad (\text{since } n \text{ is an integer}) \]
- For the upper bound:
\[ 90n < 350 \implies n < \frac{350}{90} \implies n < \frac{35}{9} \approx 3.89 \implies n \leq 3 \quad (\text{since } n \text{ is an integer}) \]

6. **List the integer values of \( n \)**:
The integer values of \( n \) that satisfy \( 1 \leq n \leq 3 \) are:
\[ n = 1, 2, 3 \]

7. **Calculate the corresponding \( x \) values**:
- For \( n = 1 \): \( x = 90 \times 1 = 90^\circ \)
- For \( n = 2 \): \( x = 90 \times 2 = 180^\circ \)
- For \( n = 3 \): \( x = 90 \times 3 = 270^\circ \)

8. **Count the number of crossings**:
The values \( 90^\circ, 180^\circ, \) and \( 270^\circ \) all lie within the interval \( (10^\circ, 350^\circ) \). Thus, the graph crosses the x-axis three times.

### Final Answer:
The graph of \( y = \sin(2x) \) crosses the x-axis **three times** in the interval \( 10^\circ \) to \( 350^\circ \).

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