How to Draw and Analyze Graphs of Trigonometric Functions with Period Amplitude and Phase Shift
To draw the trigonometry graphs of the function that are sine, cosine, and tangent, we must know the period, amplitude, phase, minimum, and maximum turning points. These graphs are useful in various fields of science and engineering.
Herewith the help of the corresponding graph, we have explained the graphical representation of sine, cosine, and tangent functions.
What are Graphs of Trigonometric Functions?
The trigonometry functions are defined by three vital trigonometry ratios: sine, cosine, and tangent. Here you will find the graph of three trigonometry functions, i.e., Sin a, Cos a, and Tan a. In the sin cos tan graph, the radian is the X-axis value of the angles and the y-axis its f(a), has the benefit of function at each given angle.
Graph of Sine (sin) Function
When the sine of an angle is placed against the angle measure, the result will be a classic ““sine curve”” shape.
To draw the sine function, mark the angle along the horizontal x-axis, and for every angle, place the sin of that angle on the vertical y-axis. The result is a smooth curve.
The curves that get this shape are known as ‘‘sinusoidal,’’ after the name of the sine function. When it appears in electronic circuits and radio, the shape is also known as a sine wave.
The Domain of Sine Function
When you drag point A around, you will notice that after a full rotation about B, the shape of the graph repeats. For each full rotation of the angle, the shape of the sine curve remains the same, and this function is called periodic. The period function is 360 degrees or 2 radians. The point can be rotated multiple times. So no matter how big you can find the sine of any angle. In terms of mathematics, the set of all real numbers is the domain of the sine function.
Range
The set of result values it produces is the range of a function. The sine function ranges from -1 to +1. When you look at the sine wave, you will notice that it never goes out of this range.
Graph of Cosine (cos) Function
When the cos of an angle is placed against the angle measurement, you get a result, which is like the classic shape of a cosine curve.
To draw the cosine function, mark the angle along the horizontal x-axis, and the cosine of that angle will be on the vertical x-axis for every angle. The result will be a smooth curve that varies from +1 to -1. It is the shape like a cosine function but is displaced to the left 90 degrees. The curves that resemble this shape are known as ‘‘sinusoidal’’ after the name of the sine function.
The Domain of Cosine Function
When you drag point A around, you will notice that after a full rotation of B, the shape of the graph repeats. The cosine curve shape is the same for each full rotation of the angle, so the function is called periodic. The period function is 360 degrees or 2 radians. You can rotate the point multiple times. According to mathematics, the domain of the cosine is the set of all real numbers.
Range
A function range is the set of result values it can produce, and the cosine function ranges from -1 to +1. The cosine curve never goes out of this range.
Few Similarities Between Sin and Cos Graph
Have curves shifted along the x-axis.
Have an amplitude of 1.
The same period of 360 degrees.
Both have the same domain that is set of all real numbers.
Graph of the Tangent (tan) Function
The Tangent of an angle is placed against that angle measure
To draw the tangent graph, mark the angle along the horizontal x-axis, and for every angle, place the tangent of that angle on the Y-axis. The result is the jagged curve that has positive infinity in one direction and negative infinity in other directions.
The Domain of the Tangent Function
It has holes in it. As you stretch the point, A around you will notice that after a full rotation of B, the shape of the graph will repeat. As the tangent curve shape is complete for each full rotation of the angle, the function is known as periodic. Here also, the function period is 360 degrees or 2 radians.
You can find the tangent of any angle with just one exception; you will see that the tan90 degree is undefined as it requires dividing by zero. Therefore, these angles are not in the domain of the tan function and give undefined results.
Range
The range of tangent function goes from positive infinity to negative infinity. As infinity is not a real number, the tan 90 degree is undefined.
FAQs on Graphs of Trigonometric Functions with Key Concepts and Visual Understanding
1. What are graphs of trigonometric functions?
Graphs of trigonometric functions are visual representations of functions like sin x, cos x, and tan x plotted against an angle (usually in radians). These graphs show how the function values change periodically as the angle increases.
- Sine and cosine graphs are smooth, wave-like curves.
- Tangent graphs have repeating vertical asymptotes.
- They are used to model periodic motion, waves, and oscillations in mathematics and physics.
2. What is the period of sine and cosine graphs?
The period of both the sine and cosine graphs is 2π radians (or 360°). The period is the length of one complete cycle of the wave.
- For y = sin x, one cycle runs from 0 to 2π.
- For y = cos x, one cycle also runs from 0 to 2π.
- If the function is y = sin(bx), then period = 2π / b.
3. How do you graph y = sin x step by step?
To graph y = sin x, plot key points over one period and draw a smooth curve through them.
- Start at (0, 0).
- At x = π/2, y = 1.
- At x = π, y = 0.
- At x = 3π/2, y = −1.
- At x = 2π, y = 0.
4. What is the amplitude of a trigonometric graph?
The amplitude of a trigonometric graph is the maximum distance from the midline to a peak or trough. For the function y = a sin x or y = a cos x, the amplitude is |a|.
- For y = sin x, amplitude = 1.
- For y = 3 sin x, amplitude = 3.
- Amplitude affects the height of the wave but not the period.
5. What is the difference between sine and cosine graphs?
The main difference between sine and cosine graphs is their starting point on the y-axis.
- y = sin x starts at (0, 0).
- y = cos x starts at (0, 1).
- Cosine is a phase shift of sine by π/2 radians.
6. What is the period of the tangent graph?
The period of the tangent graph is π radians (or 180°). This means the graph repeats every π units.
- For y = tan x, one cycle runs from −π/2 to π/2.
- If the function is y = tan(bx), then period = π / b.
- The graph has vertical asymptotes where cos x = 0.
7. What are the vertical asymptotes of the tangent graph?
The vertical asymptotes of the tangent graph occur at x = π/2 + nπ, where n is any integer. At these values, cos x = 0 and tan x is undefined.
- Examples: π/2, 3π/2, −π/2.
- The graph approaches but never touches these lines.
- These asymptotes repeat every π units.
8. How do transformations affect graphs of trigonometric functions?
Transformations change the amplitude, period, phase shift, or vertical shift of trigonometric graphs. For the function y = a sin(bx − c) + d:
- |a| = amplitude
- 2π / b = period
- c / b = phase shift
- d = vertical shift (midline)
9. What is the domain and range of sine and cosine graphs?
The domain of sine and cosine graphs is all real numbers, and their range is from −1 to 1.
- Domain: (−∞, ∞)
- Range of y = sin x: −1 ≤ y ≤ 1
- Range of y = cos x: −1 ≤ y ≤ 1
10. Why are graphs of trigonometric functions important in real life?
Graphs of trigonometric functions are important because they model periodic and oscillatory behavior in real life.
- Sound and light waves follow sine and cosine curves.
- Tides and seasonal patterns are periodic functions.
- Engineering and physics use trigonometric graphs to analyze vibrations and alternating current.
