Final answer:
In mathematics, sine functions are used to model the oscillation of waves, with amplitudes varying between +A and -A. Calculations must be done with angles in radians and checked for reasonableness, with results expressed in correct significant figures and units.
Step-by-step explanation:
To solve problems related to the sine function and sinusoidal waves in mathematics, it is important to apply the correct equations and understand the relationship between the angle and the position of the wave. The sine function oscillates between +1 and -1, which is a characteristic of all sinusoidal functions. When constructing models of waves using periodic functions, we use the sine function to express the y-position of the wave at any given point x and time t. For example, in the equation y (x, t) = A sin (kx - wt + p), A represents the amplitude, k the wave number, w the angular frequency, and p the phase shift. The value of y oscillates between +A and -A with a period corresponding to a full cycle of the wave. By substituting known values into the equation with appropriate units, radians for angles, we can obtain the y-position of the wave at any given point.
Once the calculations are done, it's important to check your answers to ensure they make sense. This might include checking if the calculated amplitude matches the expected range of oscillation and ensuring the angle unit is in radians. Additionally, using a calculator to familiarize yourself with the functions can provide valuable practice and enhance understanding.
When dealing with specific wave functions, such as y (x, t) = 4.00 cm sin (3 m¯¹x + 2) cos (4 s¯¹t), it is crucial to apply trigonometric identities and to understand the implications of the given formula, such as identifying the points of maximum displacement or antinodes.
Remember to always express your answers to problems with the correct number of significant figures and proper units, which is crucial for accuracy in scientific and mathematical calculations.