y_1=100 sin⁡(314t+40°)        y_2=70 sin⁡(314t-30°) Write the two sine waves in polar form.

To write the two sine waves in polar form, we need to express them in terms of a general polar equation for sinusoidal waves. A sinusoidal wave can be expressed in polar form as:

\[
y(t) = A \sin(\omega t + \phi)
\]

where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase angle.

### Given Equations:

1. \( y_1 = 100 \sin(314t + 40^\circ) \)
2. \( y_2 = 70 \sin(314t - 30^\circ) \)

### Conversion to Polar Form

Both equations are already in the desired polar form:

1. For \( y_1 \):
   - Amplitude (\( A_1 \)): 100
   - Angular frequency (\( \omega_1 \)): 314
   - Phase angle (\( \phi_1 \)): \( 40^\circ \)

   Thus, \( y_1 \) in polar form is:
   \[
   y_1 = 100 \sin(314t + 40^\circ)
   \]

2. For \( y_2 \):
   - Amplitude (\( A_2 \)): 70
   - Angular frequency (\( \omega_2 \)): 314
   - Phase angle (\( \phi_2 \)): \( -30^\circ \)

   Thus, \( y_2 \) in polar form is:
   \[
   y_2 = 70 \sin(314t - 30^\circ)
   \]

### Conclusion

The sine waves are already provided in polar form, as shown above. They can be represented as:

\[
y_1(t) = 100 \sin(314t + 40^\circ), \quad y_2(t) = 70 \sin(314t - 30^\circ)
\]

Would you like more details or have any questions? Here are five related questions for further exploration:

1. How would you combine \(y_1\) and \(y_2\) to form a resultant sine wave?
2. How can you express these waves using cosine instead of sine?
3. What is the phase difference between the two waves?
4. How can you determine the frequency of each wave from its angular frequency?
5. How would these waves look when plotted on a graph?

**Tip:** When converting between sine and cosine forms, use the identity \(\sin(x) = \cos(90^\circ - x)\).

y_1 = 100 sin(314t + 40°) and y_2 = 70 sin(314t - 30°). Write the two sine waves in polar form.

Learn how to express sinusoidal waves in polar form. This problem involves trigonometry, focusing on converting y_1 = 100 sin(314t + 40°) and y_2 = 70 sin(314t - 30°) into polar equations. Understand amplitude, angular frequency, and phase angle in waves. Suitable for Grades 10-12.

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