a particle moves along acurve described by y=3sin(0.5x+60degrees).state amplitude,phase angle and period

The motion of the particle is described by the equation \( y = 3\sin\left(0.5x + 60^\circ\right) \), which is a sine function. Let's break this down and extract the amplitude, phase angle, and period:

1. **Amplitude**:
   The amplitude of a sine function is the coefficient in front of the sine term. In this case, the amplitude is \( 3 \). This represents the maximum displacement from the equilibrium position.

   \[
   \text{Amplitude} = 3
   \]

2. **Phase Angle**:
   The phase angle is the constant inside the argument of the sine function that shifts the curve horizontally. In this equation, \( 0.5x + 60^\circ \) is the argument, and the phase shift is \( 60^\circ \).

   \[
   \text{Phase Angle} = 60^\circ
   \]

3. **Period**:
   The period of a sine function is given by the formula:

   \[
   \text{Period} = \frac{360^\circ}{\text{Coefficient of } x}
   \]

   In this case, the coefficient of \( x \) inside the sine function is \( 0.5 \), so the period is:

   \[
   \text{Period} = \frac{360^\circ}{0.5} = 720^\circ
   \]

Thus, the properties of the function are:
- Amplitude: 3
- Phase Angle: \( 60^\circ \)
- Period: \( 720^\circ \)

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. How does the phase angle affect the motion of the particle?
2. What is the maximum and minimum value of \( y \) during the motion?
3. How would the graph of the function change if the amplitude were doubled?
4. What is the general formula for the period of a sine function in radians?
5. How would the period change if the coefficient of \( x \) were \( 1 \) instead of \( 0.5 \)?

**Tip**: For trigonometric functions, the phase shift determines the horizontal translation, and the period determines how "stretched" or "compressed" the wave is along the x-axis.

A particle moves along a curve described by y = 3sin(0.5x + 60°). State amplitude, phase angle, and period.

Explore how to find the amplitude, phase angle, and period of the trigonometric equation y = 3sin(0.5x + 60°). Learn key concepts in trigonometry, such as sine function properties and wave motion, with step-by-step solutions. Suitable for high school students in Grades 9-12.

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