The sine and cosine functions are two of the most fundamental trigonometric functions. Their graphs provide a visual representation of how these functions vary with the angle. In this section, we will delve deeper into the characteristics of these graphs, focusing on their amplitude, period, and phase shift. For a broader understanding, you can explore the detailed analysis on graphs of sine and cosine.
Amplitude
- The amplitude of a trigonometric function represents the maximum distance it reaches from the x-axis. It gives us an idea of the height of the wave.
- For the standard sine and cosine functions, the amplitude is 1. This means that these functions oscillate between -1 and 1.
- The amplitude can be altered by multiplying the function by a coefficient. For instance, the function y = A sin(x) or y = A cos(x) has an amplitude of the absolute value of A, where A is a real number.
- The amplitude is always a positive value, regardless of the sign of A.
Example Question: Given the function y = -4 sin(x), what is its amplitude
Solution: The amplitude is the absolute value of the coefficient of the sine function, which is 4.
Period
- The period of a function is the interval over which it completes one full cycle. It gives us an idea of the width of one wave.
- For the standard sine and cosine functions, the period is 2π. This means that the function repeats its values every 2π units.
- The period can be altered by introducing a coefficient to x. For the function y = sin(Bx) or y = cos(Bx), the period P is given by P = 2π/|B|, which is elaborated in the section on solving trigonometric equations.
Example Question:Find the period of y = sin(3x).
Solution:Using the formula, P = 2π/|B|, where B = 3,The period P = 2π/3.
Phase Shift
- The phase shift represents a horizontal shift of the graph. It indicates by how much the function is shifted to the left or right.
- For the function y = sin(x - C) or y = cos(x - C), the phase shift is C. Understanding phase shifts is crucial when exploring graph transformations.
- A positive value of C shifts the graph to the right, while a negative value shifts it to the left.
Example Question: Determine the phase shift of y = sin(x + π/2).
Solution: The phase shift is -π/2 units to the left.
Graphs of Sine and Cosine
The sine and cosine functions produce wave-like curves. These curves oscillate between the values of -1 and 1. The standard graphs for these functions over one period are as follows:
- The sine function starts from 0, reaches its maximum value of 1 at π/2, returns to 0 at π, reaches its minimum value of -1 at 3π/2, and completes one cycle at 2π. To understand how these properties change with different functions, consider the graphs of tangent and cotangent.
- The cosine function, on the other hand, starts from its maximum value of 1, drops to 0 at π/2, reaches its minimum value of -1 at π, rises back to 0 at 3π/2, and completes one cycle at 2π.
Example Question:Sketch the graph of y = 2 sin(x + π/4).
Solution: Here is the graph of y = 2 sin(x + pi/4):
1. The amplitude is 2, so the function will oscillate between -2 and 2.
2. The period remains 2π as there's no coefficient for x.
3. The phase shift is -π/4 (to the left).
4. Plotting these characteristics, we get a sine curve with twice the height and shifted π/4 units to the left.
To further enrich your understanding of trigonometric functions and their applications, explore sections on compound and double angle formulas which delve into the intricacies of trigonometric identities and their proofs.
FAQ
The sine and cosine functions both represent ratios of sides in a right triangle with respect to an angle in the unit circle. As the angle varies from 0 to 2pi (or 0° to 360°), both functions complete one full cycle of their values. This is because, in the unit circle, moving 2pi radians (or 360°) brings us back to the starting point, and both sine and cosine values repeat. Hence, both functions have the same period of 2pi.
To determine the number of cycles of a sine or cosine function within a given interval, we need to consider the period of the function. The number of cycles is simply the length of the interval divided by the period of the function. For instance, if we have the function y = sin(2x) which has a period of pi, and we want to determine the number of cycles between x = 0 and x = 4pi, we would have 4pi/pi = 4 cycles. This method allows us to quickly gauge the frequency of oscillation of the function over any interval.
The sine and cosine functions are defined based on the unit circle, which has a radius of 1. The sine function represents the y-coordinate (height) of a point on the unit circle corresponding to a given angle, while the cosine function represents the x-coordinate (width). Since the radius of the unit circle is 1, the maximum distance a point can be from the x-axis (for sine) or from the y-axis (for cosine) is 1. This is why the values of sine and cosine functions are always between -1 and 1, inclusive. They can't exceed this range because they are fundamentally tied to the dimensions of the unit circle.
Vertical translations involve adding or subtracting a constant value to the entire function. For the sine and cosine functions, a vertical translation will shift the graph up or down without affecting its shape. For instance, if we have the function y = sin(x) + k, the graph of sin(x) will be shifted upwards by k units. If k is negative, it will be shifted downwards. This translation does not affect the amplitude, period, or phase shift of the function. However, it does change the maximum and minimum values of the function. For the given example, the maximum value will be 1 + k and the minimum value will be -1 + k.
The x-intercepts of the sine and cosine functions represent the angles for which the functions have a value of zero. For the sine function, this occurs when the height (or y-coordinate) of the point on the unit circle is zero. For the cosine function, it's when the width (or x-coordinate) is zero. These x-intercepts are crucial in trigonometry as they give the solutions to equations like sin(x) = 0 or cos(x) = 0. They also help in understanding the symmetry and periodic nature of these functions.
Practice Questions
For the function y = 3 sin(2x - pi):
1. The amplitude is the absolute value of the coefficient of the sine function, which is 3.
2. The period can be found using the formula P = 2pi/|B|, where B is the coefficient of x. In this case, B = 2, so the period P = pi.
3. The phase shift is given by the value inside the brackets with x. Here, it's -pi, which means the function is shifted pi units to the right.
For the function y = -2 cos(x + pi/3):
1. The amplitude is the absolute value of the coefficient of the cosine function, which is 2. However, the negative sign indicates that the graph will be reflected in the x-axis.
2. The period of the cosine function is 2pi, and since there's no coefficient for x, the period remains 2pi.
3. The phase shift is given by the value inside the brackets with x. Here, it's pi/3, which means the function is shifted pi/3 units to the left.
To sketch the graph, start by drawing the standard cosine curve. Then, reflect it in the x-axis because of the negative coefficient. Finally, shift the entire graph pi/3 units to the left to account for the phase shift. The resulting graph will have peaks at y = -2 and troughs at y = 2, and it will complete one cycle over an interval of 2pi.
