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Introduction
In trigonometry, there are six basic functions. These functions include secant, cosine, cotangent, tangent and sine. Graphs of Basic Trigonometric functions explain these functions in the form of a graph. These graphs are useful in multiple areas of science and engineering, ranging from engines and waves to the growth of plants and animals.
The trigonometric identities and functions are ratios of sides of a right-angled triangle and tend to be used extensively in algebra, geometry and calculus. The sides of a right triangle include the base, hypotenuse and perpendicular side, all of which are used to calculate values of trigonometry functions with the usage of the relevant trigonometry formula. Conversely, to sketch the trigonometry graphs of the functions, you will need to know the maximum and minimum turning points, along with a period, amplitude and phase.
The Unit circle
A circle whose radius equals 1 is a unit circle. The angle θ is formed from a ray that extends from origin through a point p on the x-axis and unit circle. The value of cos θ equals to x- coordinate of point p and the value of sin θ equals to y-coordinate of point p.
The unit circle shows measurements of angles in both degrees and radians. Start at 0π, and then follow the circle counterclockwise. When the angle θ increases to 90° or π/2 radians, the value of cosine goes down as the point approaches the y-axis. On the other hand, the value of sine goes up. The point moves 2π or 360° when a single counter-clockwise revolution is completed.
Graphing Sine and Cosine Functions
For graphs of cosine and sine functions, use a unit circle to make the graph for y = cos x and y = sin x and then evaluate the function
You need to set up a table of values with the usage of intervals 0π, π/2, 3π/ 2 and 2π for x and calculate the consequent y value.
f(x) or y = sin x
f(x) or y | x |
0 | 0π, |
1 | π/2 |
0 | π |
-1 | 3π/ 2 |
0 | 2π |
Label x-axis with values 0π, π/2, π, 3π/2 and 2π for drawing the graph of a single period of sine or y = sin x. After doing so, plot points for the value of y or f(x) from the circle unit or the table.
Other points can be added for intermediate values between the ones mentioned above to get a more complete graph. A line could be drawn by choosing to connect the points while showing 1 and a half periods of the sine function.
You can graph cosine function y = cos x like sine function, and use the table of values given below:
f(x) or y = cos x
f(x) or y | x |
0 | 0π, |
1 | π/2 |
0 | π |
-1 | 3π/ 2 |
0 | 2π |
For drawing a graph of a single period of cosine or y = cos x, you would be required to label the x-axis with values 0π,/2, π,3𝜋/2, and 2π. After doing so, you can plot points for the value of y or f(x) from either the unit circle or the table.
Other points can be added for in-between values between the ones mentioned above to get a more absolute graph.
Graphing Tangent Function y = tan x
The valuation of the tangent at 0π for the unit circle is 0/1. This is equal to 0. The value of tangent at 𝜋/2 is 1/0, and it will yield a divide by 0 error or not defined as per a trigonometry calculator. Hence, a tangent function is not defined at 𝜋/ 2, which is presented by drawing asymptote at 𝜋/2.
The value of the tangent at π is 0/1, and hence it results in 0. To determine how the tangent tends to behave amid 0π and asymptote, you need to identify the sine and cosine values of π/4. This is halfway between 0π and π/2. The tangent of π/4 is √2/2(sine) divided by √2/2 (cosine). You will get √2/2×2/√2 which simplifies to 1 by flipping the cosine value and multiplying.
Value of the tangent at π/4 hence is one. The tangent of 3𝜋/4 is √2/2 (sine) divided by –√2/2 (cosine) for calculating the value of tangent for 3𝜋/ 4, as per the trigonometry formula. This shall simplify to a tangent value of -1.
Conclusion
Graphs of Basic Trigonometric functions have a domain value of θ represented on the horizontal x-axis. Their range value additionally is represented along the vertical y-axis. Graphs of tanθ and sinθ pass through the origin, while graphs of other trigonometric functions do not do so. Sinθ and Cosθ range is limited to [-1, 1]. These graphs come as a huge help in multiple real-life applications, especially in the domain of engineering.
