Sine Function
The sine function, symbolised as sin, is a quintessential function in trigonometry, defined as the ratio of the length of the side opposite to an angle in a right-angled triangle to the length of the hypotenuse.
Graph of Sine Function
The graph of the sine function, y = sin x, exhibits a wave-like pattern, oscillating between -1 and 1 on the y-axis. This wave repeats every 2π radians (or 360 degrees), creating a periodic pattern across the x-axis.
Key Characteristics
- Amplitude: The maximum positive displacement from the x-axis, which is 1 for the standard sine function.
- Period: The length for one complete cycle of the wave, which is 2π for the standard sine function.
- Frequency: The number of cycles completed in a unit interval, which is the reciprocal of the period.
- Phase Shift: The horizontal displacement of the graph, which is zero for the standard sine function.
Transformations
To delve deeper into the manipulation of the sine function's graph, explore our notes on function transformations.
- Vertical Stretch/Compression and Reflection: y = a sin x modifies the amplitude and may reflect the graph across the x-axis if a < 0.
- Horizontal Stretch/Compression and Reflection: y = sin bx affects the period and may reflect the graph across the y-axis if b < 0.
- Vertical Translation: y = sin x + d shifts the graph up or down by d units.
- Horizontal Translation: y = sin(x + c) shifts the graph left or right by c units.
Example Question 1: Graph Analysis
Analyse the graph of y = 2sin(3x).
- Amplitude: 2
- Period: 2π/3
To sketch the graph, identify the points where it crosses the x-axis, reaches its peak, and trough, and use these to draw one complete wave. Then, extend the pattern to complete the graph.
Cosine Function
The cosine function, denoted as cos, is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It shares several properties with the sine function but with a phase shift of π/2.
Understanding the relationship between the sine and cosine functions is fundamental when studying trigonometric identities.
Graph of Cosine Function
The graph of the cosine function, y = cos x, also exhibits a wave-like pattern, similar to the sine function but with a different starting point. The graph of the cosine function starts at its maximum, while the sine function starts at zero.
Key Characteristics
- Amplitude: 1 for the standard cosine function.
- Period: 2π for the standard cosine function.
- Frequency: The reciprocal of the period.
- Phase Shift: π/2 to the left compared to the sine function.
Transformations
- Vertical Stretch/Compression and Reflection: y = a cos x modifies the amplitude and may reflect the graph across the x-axis if a < 0.
- Horizontal Stretch/Compression and Reflection: y = cos bx affects the period and may reflect the graph across the y-axis if b < 0.
- Vertical Translation: y = cos x + d shifts the graph up or down by d units.
- Horizontal Translation: y = cos(x + c) shifts the graph left or right by c units.
Example Question 2: Graph Transformation
Determine the amplitude and period of the graph y = -4cos(x/2).
- Amplitude: 4
- Period: 4π
Tangent Function
The tangent function, denoted as tan, is the ratio of the sine to the cosine of an angle: tan x = sin x / cos x. Unlike sine and cosine, the tangent function does not have a constant amplitude and can take values from -∞ to ∞.
For more complex examples involving the tangent function, refer to solving trigonometric equations.
Graph of Tangent Function
The graph of y = tan x has a series of vertical asymptotes, which are lines that the graph approaches but never touches, at x = (2n + 1)π/2. The period of the tangent function is π.
Key Characteristics
- Amplitude: Not defined due to the asymptotes.
- Period: π for the standard tangent function.
- Frequency: The reciprocal of the period.
- Phase Shift: None for the standard tangent function.
Transformations
- Vertical Stretch/Compression and Reflection: y = a tan x modifies the steepness and may reflect the graph across the x-axis if a < 0.
- Horizontal Stretch/Compression and Reflection: y = tan bx affects the period and may reflect the graph across the y-axis if b < 0.
- Vertical Translation: y = tan x + d shifts the graph up or down by d units.
- Horizontal Translation: y = tan(x + c) shifts the graph left or right by c units.
Example Question 3: Finding the Period
Find the period of the graph y = tan(2x).
- Period: π/2
Applications in Real Life
Circular functions are not just theoretical constructs but have practical applications in various fields such as physics, engineering, and computer science. For instance:
- Physics: Describing oscillations, waveforms, and circular motion. The integration of trigonometric functions is particularly useful in physics for solving problems involving motion.
- Engineering: Analysing waveforms in signal processing and electrical circuits. Engineers often rely on the differentiation of trigonometric functions to model and understand the behavior of systems.
- Computer Science: Generating computer graphics, particularly in animations and gaming. Understanding how to apply function transformations can be crucial for creating realistic animations.
Practice Questions
1. Graph Sketching: Sketch the graph of y = 3sin(2x) and identify the amplitude and period. This task is an excellent way to apply your understanding of function transformations.
2. Solving Equations: Solve for x in the equation 2cos x - 1 = 0 for 0 ≤ x ≤ 2π. For more complex trigonometric equations, refer to solving trigonometric equations.
3. Application: A ferris wheel has a diameter of 60m and rotates once every 10 minutes. Write a sine function to model the height of a point on the edge of the wheel above the ground over time. This real-world scenario exemplifies the applications of trigonometric functions in engineering and physics.
FAQ
The graphs of sine and cosine functions continue indefinitely because these functions are periodic and oscillate between their maximum and minimum values without bounds. The concept of periodicity implies that the function repeats its values in regular intervals or periods, and for sine and cosine functions, this interval is 2pi. The endless continuation of the sine and cosine graphs represents the constant cyclical nature of these functions. In real-world contexts, this infinite behaviour models phenomena that are inherently periodic, such as the rotation of a wheel, the oscillation of a pendulum, or the alternating current in electrical circuits, which continue to oscillate back and forth indefinitely under ideal conditions.
The frequency of a sine or cosine function is determined by how many complete cycles the function undergoes in a given interval. Mathematically, frequency is the reciprocal of the period of the function. To determine the frequency from the graph, first, identify the period, which is the horizontal length of one complete cycle of the graph. The period can be found by identifying the distance between two consecutive peaks or troughs on the graph. Once the period (T) is identified, the frequency (f) can be found using the relation f = 1/T. The frequency is crucial in applications like signal processing and telecommunications, where understanding the rate at which a signal oscillates is paramount for information transmission and modulation.
Yes, the tangent function can be expressed in terms of sine and cosine functions. The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side. In trigonometric function terms, tan(theta) is expressed as the ratio of sin(theta) to cos(theta), mathematically written as tan(theta) = sin(theta)/cos(theta). This relationship is fundamental in trigonometry and is used to derive various identities and to solve trigonometric equations. Understanding this relationship is also crucial in calculus, particularly when dealing with derivatives and integrals involving trigonometric functions, and in physics, where it helps to resolve vectors and analyse motion.
The amplitude of sine and cosine functions, which is the maximum positive displacement from the midline, plays a pivotal role in real-world applications, particularly in oscillatory phenomena. For instance, in physics, the amplitude of a sine or cosine wave could represent the maximum displacement of a particle from its equilibrium position in wave mechanics or simple harmonic motion. In electrical engineering, it could denote the maximum voltage in an alternating current (AC) circuit. The amplitude provides crucial information about the extent of variation from the mean value, which is vital for understanding and predicting the behaviour of physical systems, such as predicting the maximum tide height in oceanography or determining the intensity of a sound wave in acoustics.
The unit circle, which is a circle with a radius of 1 centred at the origin of a coordinate plane, is intrinsically linked to the sine and cosine functions. When an angle is formed by rotating a radius of the unit circle about the origin, the x and y coordinates of the point where the radius intersects the circle correspond to the cosine and sine of that angle, respectively. Essentially, for an angle theta in standard position (vertex at the origin and initial side along the positive x-axis), cos(theta) gives the x-coordinate and sin(theta) gives the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This relationship is fundamental in trigonometry and provides a geometric interpretation of these functions, especially when considering their periodic nature and the generation of their respective graphs.
Practice Questions
Given the function y = 3sin(2x - pi) + 1, sketch the graph for 0 <= x <= 2pi and identify the amplitude, period, phase shift, and vertical translation.
The amplitude of the function y = 3sin(2x - pi) + 1 is 3, as it is the coefficient of the sine function. The period is pi because the coefficient of x inside the sine function is 2, and the period of sine is 2pi, so we divide 2pi by the coefficient to get pi. The phase shift is pi/2 to the right, as the term inside the sine function is subtracted by pi, and we divide by the coefficient of x to find the actual shift. The vertical translation is 1 unit upwards, as indicated by the "+ 1" outside of the sine function. To sketch the graph, we can start by plotting the midline at y = 1 and then sketching a sine wave that reaches from -2 to 4 on the y-axis, ensuring that it completes one full cycle from x = 0 to x = pi and another from x = pi to x = 2pi.
Solve the equation 2cos2(x) - 3cos(x) - 5 = 0 for 0 <= x <= 2pi.
To solve the trigonometric equation 2cos2(x) - 3cos(x) - 5 = 0, we can use a substitution method. Let's let u = cos(x). Substituting this into the equation, we get a quadratic equation in terms of u: 2u2 - 3u - 5 = 0. Factoring the quadratic equation, we get (2u + 1)(u - 5) = 0. Setting each factor equal to zero and solving for u, we find u = -1/2 and u = 5. However, since the cosine function only takes values between -1 and 1, u = 5 is not a valid solution. So, we only consider u = -1/2. Substituting back, we have cos(x) = -1/2. To find the values of x that satisfy this equation in the given interval, we consider the cosine function. x could be 2pi/3 or 4pi/3 in the interval 0 <= x <= 2pi. So, the solutions to the equation 2cos2(x) - 3cos(x) - 5 = 0 are x = 2pi/3 and x = 4pi/3.
