Cosine function

cos(x), cosine function.

Although the sine function is probably the most familiar of the six modern trigonometric functions, the cosine function comes a close second. Indeed, the sine and cosine functions are very closely related, as we shall see (if you are not familiar with the sine function, you may wish to read the page entitled "The Sine Function"). However, whereas the modern sine function is closely related to the chord function used by the ancient Greeks, the cosine function has no equivalent in Greek trigonometry. In fact, the Greeks had no other trigonometric functions apart from the chord.

Nevertheless, they were aware of the association between the sine and the cosine functions, albeit not in a trigonometric sense. The way in which the two functions relate to each other is captured in geometric terms by the Pythagorean theorem, which defines the relationship between the sides of a right-angled triangle. Consider the illustration below.


The relationship between sine and cosine

The relationship between sine and cosine


Cosine definition

In a right triangle ABC the sine of α, sin(α) is defined as the ratio between the side adjacent to angle α and the side opposite to the right angle (hypotenuse):

cos α = b / c

Example

b = 3"

c = 5"

cos α = b / c = 3 / 5 = 0.6

If you ask a calculator to find the arc cosine (cos-1 or acos) of a number, it cannot return an infinitely long list of angles, so by convention it returns just the first one. But remember, there are many others.

Graph of cosine

2.1: Graphs of the Cosine and Sine Functions - Mathematics LibreTexts

To graph the cosine function, we mark the angle along the horizontal x axis, and for each angle, we put the cosine of that angle on the vertical y-axis. The result, as seen above, is a smooth curve that varies from +1 to -1. It is the same shape as the cosine function but displaced to the left 90°. Curves that follow this shape are called 'sinusoidal' after the name of the sine function whose shape it resembles. 

In the diagram above, drag the point A around in a circular path to vary the angle CAB. As you do so, the point on the graph moves to correspond with the angle and its cosine. (If you check the "progressive mode" box, the curve will be drawn as you move the point A instead of tracing the existing curve.)

The domain of the cosine function

As you drag the point A around notice that after a full rotation about B, the graph shape repeats. The shape of the cosine curve is the same for each full rotation of the angle and so the function is called 'periodic'. The period of the function is 360° or 2π radians. You can rotate the point as many times as you like. This means you can find the cosine of any angle, no matter how large. In mathematical terms we say the 'domain' of the cosine function is the set of all real numbers.

Range

The range of a function is the set of result values it can produce. The cosine function has a range that goes from -1 to +1. Looking at the cosine curve you can see it never goes outside this range.


Cosine rules

Rule name Rule
Symmetry cos(-θ) = cos θ
Symmetry cos(90°- θ) = sin θ
Pythagorean identity sin2(α) + cos2(α) = 1
  cos θ = sin θ / tan θ
  cos θ = 1 / sec θ
Double angle cos 2θ = cos2 θ - sin2 θ
Angles sum cos(α+β) = cos α cos β - sin α sin β
Angles difference cos(α-β) = cos α cos β + sin α sin β
Sum to product cos α + cos β = 2 cos [(α+β)/2] cos [(α-β)/2]
Difference to product cos α - cos β = - 2 sin [(α+β)/2] sin [(α-β)/2]
Law of cosines  
Derivative cos' x = - sin x
Integral ∫ cos x dx = sin x + C
Euler's formula cos x = (eix + e-ix) / 2

Inverse cosine function

The arccosine of x is defined as the inverse cosine function of x when -1≤x≤1.

When the cosine of y is equal to x:

cos y = x

Then the arccosine of x is equal to the inverse cosine function of x, which is equal to y:

arccos x = cos-1 x = y

Example

arccos 1 = cos-1 1 = 0 rad = 0°

See: Arccos function

The graph of the inverse cosine function is a lot like the graph of the inverse sine function. They're tight like that. It needs to be chopped into pieces, too, in order to pass the vertical line test. Poor inverse trig functions. They never get to be themselves.

To graph the inverse cosine function, we limit or choose a section of our regular cosine graph to work with. Here's y = cos x.

Next, we limit the domain to [0°, 180°], or [0, π] if you're feeling radian-y.

To find the inverse cosine graph, we have to swap the variables, which gives us:

Inverse cos has a domain of [-1, 1] and a range of [0°, 180°], or [0, π].

That limits us to Quadrant I and Quadrant II when we're solving problems with the inverse cosine function.


Cosine table

x

(°)

x

(rad)

cos x
180° π -1
150° 5π/6 -√3/2
135° 3π/4 -√2/2
120° 2π/3 -1/2
90° π/2 0
60° π/3 1/2
45° π/4 2/2
30° π/6 3/2
0 1

And now the main question. When do we use Cosine function?

Cos function is often used in excel as a mathematical function that calculates the Cos value of any angle. As we know, Cos angle is the ratio of Base to Hypotenuse, but in excel, we need to select radians function here along with COS Function to get the proper angle value. Also, we can multiply the input angel value with Pi()/180 other than using the Radians function.


The cosine function, along with sine and tangent, is one of the three most common trigonometric functions. In any right triangle, the cosine of an angle is the length of the adjacent side (A) divided by the length of the hypotenuse (H). In a formula, it is written simply as 'cos'.


cos =


A
H

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