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Cosine Curve

The graph corresponding to the function y = f(x) = cos(x) is often referred to as the cosine curve. In trigonometry three functions (sin, cos, tan) are used for finding the angles and length of the sides of the triangle. The graph of cosine curve is also used for the finding the area under the curves. In this article, we analyze the basic properties and graph of cosine function.

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Cosine Curve Graph

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Cosine curve, is a smooth curve that varies from +1 to -1. It is the same shape as the sine function except that, this curve cuts the y-axis at 1 whereas sine curve passing through the origin. The shape of the cosine curve is the same for each complete rotation of the angle and so the function is called periodic. The period of the function is 360o or 2$\pi$ radians.

The equation of cosine function:


y = cos x

Domain and range of cosine function

Domain = $(-\infty, +\infty)$

Range = [-1, 1]

Cosine Curve


  • The total period of cosine function is 2$\pi$.
  • cosine curve starts at 1 and goes down to 0.
  • The cosine curve look like a sine curve except that it is translated.

Cosine Curve Equation

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The general form of the cosine function:

Y = a cos(x)
Here, a is the amplitude of the curve
and the period of this cosine curve is 2$\pi$.

So, the cosine curve looks as follows:

Cosine Curve Equation

Sine and Cosine Curves

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A sine and cosine curves are graphic representation of the sine and cosine function. The shape of the sine and cosine curve is the same for each complete rotation of the angle and so the function is called periodic. The graph of a function is called the collection of all terms (x, y) coordinates. Sine and cosine graphs show a repeated pattern that occurs every 2$\pi$ or 360o .

Cosine Curve Examples

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Given below are some of the examples on Cosine curve.

Solved Examples

Question 1: Draw the graph for the function, y = 3 cos t.
Solution:
The given curve looks like the general format of cosine with the amplitude of 3.Both the positive and negative amplitude value is 3. So, in the graph, the amplitude a value is both 3 on positive and the negative on the graph.

The period of the cosine function is 2 $\pi$

Cosine Curve Examples

Question 2: Draw the graph of the following function y = a cos(x). For a = 1, 1.5, 4, 0.8, 0.1
Solution:
Here, a = 1, 1.5, 4, 0.8, 0.1

Cosine Curve Example

Question 3: Draw the graph for the function, Y = 10 cos 3x.
Solution:

Given function is Y = 10 cos 3x.

Step 1:
Here, the amplitude of the cosine function is 10

And, the Period of the cosine function is $\frac{2 \times \pi}{b}$

This is the general formula for finding the period of the cosine functions. Here, the value of 2$\pi$ is the fixed time interval because the total time interval of cosine function is 2$\pi$. So, it cannot be changed for any of the cosine function.

So, the period of the given cosine curve, x = $\frac{2 \times \pi}{3}$

Step 2:
Based on the above two information, we can draw the graph of the cosine curve. So, the below diagram shows the graph the cosine functions of y = 10 cos 3x.

Cosine Curve Graph



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