How do you know where to plot the points of a cosine function on a graph? Is there a more simple way then plugging the x value into the calculator?
nope.
Although you do know that the period is 2π, so it crosses the x-axis at intervals of π, with
cos(0) = 1
cos(π/2) = 0
cos(π) = -1
cos(3π/2) = 0
cos(2π) = 1
Then sketch a smooth curve using those points. If you want other points, you can use known values at π/6, π/4, π/3 and their multiples.
Most other points you will need to use your calculator.
To plot points of a cosine function on a graph, you can use the values of the function at specific angles or x-values. While you can use a calculator to find the cosine values for different x-values, there are alternative methods that can simplify the process.
One helpful concept is the unit circle. The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) of a coordinate plane. By referencing the angles in the unit circle, you can determine the cosine values without a calculator.
Here's a step-by-step process to plot points of a cosine function on a graph without relying solely on a calculator:
1. Determine the period of the cosine function: The period is the distance between two consecutive peaks or troughs of the graph. For a standard cosine function with no transformations, the period is 2π.
2. Divide the period into equal intervals: Divide the period into equal intervals depending on the desired precision. For example, if you want ten points, divide the period into ten sections.
3. Calculate the angle for each interval: Divide the period by the number of intervals to find the increment angle. In this case, 2π divided by ten would give you π/5.
4. Use the unit circle to find the cosine value: Start with an angle of 0, then add the increment angle to the previous angle for each interval. For each angle, locate the corresponding point on the unit circle and read the x-coordinate, which is the cosine value.
5. Plot the points on the graph: Transfer the angle and cosine value to the graph paper or software. The angle corresponds to the x-coordinate, and the cosine value corresponds to the y-coordinate.
By following these steps, you can obtain the points of the cosine function on the graph without solely relying on a calculator. This method can be helpful for understanding the pattern of the function and developing a visual intuition for cosine graphs.