Find the period of the following fuctions:
(i) sin2x (ii)cos3x (iii) tan2x

Answer

To find the period of the given functions \( \sin 2x \), \( \cos 3x \), and \( \tan 2x \), we will use the standard formulas for the periods of sine, cosine, and tangent functions.

### Step-by-Step Solution:

1. **Finding the Period of \( \sin 2x \)**:
- The standard form of the sine function is \( y = A \sin(Bx + C) \).
- Here, \( A = 1 \), \( B = 2 \), and \( C = 0 \).
- The period \( T \) of the sine function is given by the formula:
\[ T = \frac{2\pi}{B} \]
- Substituting \( B = 2 \):
\[ T = \frac{2\pi}{2} = \pi \]
- Therefore, the period of \( \sin 2x \) is \( \pi \).

2. **Finding the Period of \( \cos 3x \)**:
- The standard form of the cosine function is \( y = A \cos(Bx + C) \).
- Here, \( A = 1 \), \( B = 3 \), and \( C = 0 \).
- The period \( T \) of the cosine function is also given by:
\[ T = \frac{2\pi}{B} \]
- Substituting \( B = 3 \):
\[ T = \frac{2\pi}{3} \]
- Therefore, the period of \( \cos 3x \) is \( \frac{2\pi}{3} \).

3. **Finding the Period of \( \tan 2x \)**:
- The standard form of the tangent function is \( y = A \tan(Bx + C) \).
- Here, \( A = 1 \), \( B = 2 \), and \( C = 0 \).
- The period \( T \) of the tangent function is given by:
\[ T = \frac{\pi}{B} \]
- Substituting \( B = 2 \):
\[ T = \frac{\pi}{2} \]
- Therefore, the period of \( \tan 2x \) is \( \frac{\pi}{2} \).

### Summary of Results:
- The period of \( \sin 2x \) is \( \pi \).
- The period of \( \cos 3x \) is \( \frac{2\pi}{3} \).
- The period of \( \tan 2x \) is \( \frac{\pi}{2} \).