All trignometric functions are periodic but only sine or cosine functions are used to define SHM. Why?

Options

A. Pythgoras
B. Carl Friedrich Gauss
C. Leonhard Euler
D. Jean Baptiste Joseph Fourier

Answer

### Step-by-Step Solution

1. **Understanding Simple Harmonic Motion (SHM)**:
- Simple Harmonic Motion is defined as a type of periodic motion where the restoring force acting on the object is directly proportional to its displacement from the mean position and is directed towards that mean position.
- In SHM, there are two extreme positions (maximum displacements) and one mean position.

2. **Defining Amplitude**:
- The maximum displacement from the mean position is called the amplitude (denoted as A). In SHM, the displacement (X) varies between -A (left extreme position) and +A (right extreme position).

3. **Periodic Nature of Trigonometric Functions**:
- All trigonometric functions are periodic, meaning they repeat their values in regular intervals. However, the range of these functions varies.

4. **Range of Sine and Cosine Functions**:
- The sine (sin) and cosine (cos) functions have a range of values between -1 and +1. This means that when we use these functions to describe displacement in SHM, the displacement will always remain within the bounds of -A and +A.

5. **Displacement Equations in SHM**:
- The displacement in SHM can be expressed using the equations:
- \( X = A \sin(\omega t) \)
- \( X = A \cos(\omega t) \)
- Since the sine and cosine functions only take values between -1 and +1, the displacement X will also remain bounded between -A and +A.

6. **Other Trigonometric Functions**:
- Other trigonometric functions such as tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) do not have a bounded range. They can take values from 0 to infinity or can be undefined, which makes them unsuitable for defining SHM.

7. **Conclusion**:
- Since SHM is a bounded motion, only the sine and cosine functions are appropriate to represent it, as they ensure that the displacement remains within the defined limits of -A and +A.