Key Features of the Sine Graph
- Amplitude: The amplitude of the sine graph is 1. This means the maximum height (or depth) of the graph from the x-axis is 1 unit. The graph oscillates between -1 and 1.
- Period: The period of the sine graph is 2π radians or 360°. This means the graph completes one full cycle in 2π radians. For more on radians, refer to the introduction to radians.
- Frequency: The frequency is the number of cycles the graph completes in 2π radians. For the standard sine graph, the frequency is 1.
- Phase Shift: The standard sine graph starts from the origin (0,0). If the graph is shifted to the left or right, it is said to have a phase shift.
- Vertical Shift: If the entire graph is moved up or down, it is called a vertical shift.
Transformations of the Sine Graph
Transformations can change the appearance of the sine graph. The general form of a transformed sine function is: y = a sin(bx + c) + d. You can learn more about basic transformations and how they affect other functions.
Where:
- a is the amplitude. If a is negative, the graph is reflected in the x-axis.
- b affects the period of the graph. The period is given by 2π/|b|.
- c is the phase shift. The graph shifts c/b units horizontally.
- d is the vertical shift. The graph shifts d units vertically.
Example 1: Graphing a Transformed Sine Function
Consider the function y = 2 sin(x + π/2) - 1.
- The amplitude is 2.
- The period remains 2π as there's no coefficient with x.
- The phase shift is -π/2 (shifted to the left by π/2 units).
- The vertical shift is -1 (shifted down by 1 unit).
To graph this function, start by plotting the standard sine curve. Then, stretch it vertically by a factor of 2, shift it to the left by π/2 units, and finally, move it down by 1 unit. For more on graphing, you can also see graphs of tangent.
Example 2: Solving an Equation Using the Sine Graph
Solve for x in the equation sin(x) = 0.5 for 0 <= x <= 2π.
Using the sine graph, we can see that sin(x) = 0.5 at two points in the interval [0, 2π]. These points are x = π/6 and x = 5π/6. These points are x = π/6 and x = 5π/6. Understanding the arc length can also aid in solving such equations.
Deeper Dive into the Sine Function
The sine function can also be represented in various forms:
- Alternate Form: sin(x) = 1/2 i e(-i x) - 1/2 i e(i x)
- Roots: The sine function is zero at integer multiples of π, i.e., x = π n where n is an integer.
- Parity: The sine function is odd, which means sin(-x) = -sin(x).
- Series Expansion: The Taylor series expansion of the sine function around x = 0 is x - x3/6 + x5/120 + ...
- Derivative: The derivative of sin(x) is cos(x).
- Integral: The indefinite integral of sin(x) is -cos(x) + constant. For more advanced topics, see double and half-angle identities.
Applications in Real Life
The sine graph is not just a mathematical concept; it has real-world applications. For instance, it can represent phenomena that oscillate, such as sound waves, light waves, and the tides of the ocean. By understanding the key features and transformations of the sine graph, students can better analyse and interpret such phenomena in various scientific fields. More examples of transformations are discussed in the basic transformations section.
FAQ
The sine function is intrinsically linked to the unit circle, which is a circle with a radius of 1 unit centred at the origin of a coordinate plane. When we rotate a point on the unit circle about the origin, the y-coordinate of that point gives the sine value of the angle of rotation. In other words, if you draw a radius on the unit circle that makes an angle with the positive x-axis, the y-coordinate of the point where the radius intersects the circle is the sine of that angle. This relationship provides a geometric interpretation of the sine function and forms the basis for understanding its properties and behaviour.
The oscillatory nature of the sine function makes it an ideal mathematical tool for representing phenomena that exhibit wave-like or cyclic behaviour. In physics, the sine function is used to describe simple harmonic motion, such as the movement of a pendulum or a spring. In engineering, it's used to analyse and design electrical circuits, especially those involving alternating current (AC). In music, sound waves can be represented using sine waves, with different pitches corresponding to different frequencies. In astronomy, the sine function helps in understanding the periodic motion of celestial bodies. The repetitive and predictable nature of the sine function makes it invaluable in these and many other real-world applications.
While both the sine and cosine functions are fundamental trigonometric functions and share many similarities, their graphs have distinct differences. The primary difference lies in their starting points. The sine graph starts from the origin (0,0), whereas the cosine graph starts from the point (0,1). This means that the cosine graph is a horizontal shift of the sine graph by π/2 units to the left. Another way to view this is that the cosine function is the sine function with a phase shift of π/2. Despite these differences, both graphs have the same amplitude, period, and shape, and they both oscillate between -1 and 1.
No, the sine function cannot take values outside the range [-1, 1]. Regardless of the input value or angle, the output of the sine function will always lie between -1 and 1, inclusive. This is because the sine function represents the y-coordinate of a point on the unit circle, and since the radius of the unit circle is 1, the y-coordinate can never exceed 1 or be less than -1. This range is a defining characteristic of the sine function and is evident when you look at its graph, which oscillates between these two values.
The sine function is considered periodic because it repeats its values in regular intervals or periods. For the standard sine function, this interval is 2π radians or 360°. This means that for every 2π radians, the values of the sine function will start to repeat in the same sequence. This periodic nature is evident when you look at the graph of the sine function, where you can see the wave-like pattern repeating itself. This periodicity is a fundamental characteristic of trigonometric functions and plays a crucial role in their applications, especially in fields like physics, engineering, and music, where wave patterns and oscillations are common.
Practice Questions
The given function is y = 3 sin(2x - pi) + 4.
- The amplitude is the coefficient of the sine function, which is 3.
- The coefficient of x inside the sine function is 2, so the period is 2pi/2 = pi.
- The phase shift can be determined from the term inside the sine function. The graph is shifted to the right by pi/2 units.
- The vertical shift is given by the constant term, which is 4, meaning the graph is shifted up by 4 units.
To sketch the graph, start with the basic sine curve. Then, stretch it vertically by a factor of 3, compress it horizontally by a factor of 2, shift it to the right by pi/2 units, and finally, move it up by 4 units.
The given function is y = 5 sin(3x + pi/3).
- The amplitude of the sound wave is the coefficient of the sine function, which is 5.
- The phase shift can be determined from the term inside the sine function. The graph is shifted to the left by pi/9 units.
To find the value of x for which the sound wave reaches its first maximum after x = 0, we need to find the first positive value of x for which the argument of the sine function is pi/2. Setting 3x + pi/3 = pi/2, we get x = pi/6 - pi/9 = pi/18. Thus, the sound wave reaches its first maximum at x = pi/18.
