Cosine Functions

The ratio between the adjacent side of the right-angled triangle and the hypotenuse of the right-angled triangle is called the cosine function. The cosine function is included in the three main primary trigonometry functions and it is the complement of the sine function, which is cosine. There are various laws of cosines in trigonometry which makes it a tough topic for students across boards.

The hypotenuse is regarded as the longest side of a right-angle triangle. The side adjacent to it is called the height of the right-angle triangle. The side adjacent to the height is called the base of the right-angled triangle. The ABC is a triangle with AB regarded as the hypotenuse and the angular vertex hence created will be between AB and BC. The angle between the two is named as α.

On a side note, the way to define cosine in maths is tricky for many, but once you understand the concept things tend to get easier. Though the process of teaching cosine needs some updates.  

Cosine Formula

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By the definition of cosine, the knowledge is now gathered that it is the adjacent side divided by the hypotenuse. hence, from the above diagram,

cos α = AC/AB

else

cos α = b/h

Cosine Table

Cosine Degrees

Values to be used

Cos 0°

(+)1

Cos 30°

(+) \[\frac{{\sqrt{3}}}{(2)}\]

Cos 45°

(+) \[\frac{{1}}{(\sqrt{2})}\]

Cos 60°

(+) \[\frac{{1}}{(2)}\]

Cos 90°

Zero

Cos 120°

(-) \[\frac{{1}}{(2)}\]

Cos 150°

(-) \[\frac{{\sqrt{3}}}{(2)}\]

Cos 180°

(-)1

Cos 270°

Zero

Cos 360°

(+)1

 

The table above shows not only the basic additional cos values, but we have a bit further to include additional cos value- Enjoy the learning.

Now, if we talk from the cartesian point of view, it is legit fun to know that the cosine function values change concerning the quadrants. In the table provided above, we can note that at 120,150,180 degrees, the values of cosine functions have negative values concerning their angular measurements. While the values of the cosine function for the angles 0,30, etc. have positive values. Now, it is just because they are present at a quadrant structure of the graph. If we plot the cosine graph, we can easily find out these values concerning the angles. As far as cosine functions are concerned, we will see that,

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Cosine Properties with Respect to the Quadrant

Range of degrees

Quadrant

Cosine Function Sign

Cosine Value Range

0 to 90 Degrees

First Quadrant

+ (Positive)

0 < cos(x) < (1)

90 to 180 Degrees

Second Quadrant

– (Negative)

(-1) < cos(x) < (0)

180 to 270 Degrees

Third Quadrant

– (Negative)

(-1) < cos(x) < (0)

270 to 360 Degrees

Fourth Quadrant

+ (Positive)

0 < cos(x) (<10)

 

Cosine Function Graph 

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Just as the characters of the sine graph, the cosine function graph also goes back and forth of the x-axis as the sine function graph does. There is a subtle difference between the two that is the sine function graph starts from the 0 positions, but the cosine function graph starts from the area of 90 degrees. The cosine graph given above starts from the one spot and then falls to the -1 place. After that, it keeps rising again. This rising and dipping pattern of a graph of the cosine function graph allows us to witness a clear picture of the behaviour of the cosine function in the cartesian coordinates.

Inverse Cosine Functions

The inverse cosine functions are also known as the arc cosine functions. These functions are generally used for the measurement of any angle of the right-angled triangle where the ratio of adjacent side and hypotenuse is provided. It is written as Arcos or cosine-1.

For a right triangle with sides A, B, and C, the cos function can be used to measure the angle. By cos-1(AB/BC), cos-1(BC/AC),), cos-1(AC/AB).

Properties Of Cosine Function

 Cosine Identities which are to be paid attention to:

  • cos2 (g) + sin2 (g) = 1

  • cos θ = (1)/sec θ

  • cos (−) θ = cos (θ)

  • art cos cos (x)) = x + 2zπ  [where z=integer]

  • Cos (2y) = cos2 (y) − sin2 (y)

  • cos (θ) = (sin) (π/2 − θ)

Cosine function uses in Calculus

For cosine function f(g) = cos(g), the differentiation and the integration can be given as:

Derivative of cos(g), f′ (g) = −sin (g)

Integral of cos(g), ∫f (g) dx = −sin(g) + Constant term of the integral.

Law of Cosines in Trigonometry

There is a relation between the side and the angles of the triangle. If a triangle with the sides h, k, g is taken with angles H, K, L, the cosine rule would be applied as the equation provided below. 

Discussion of the cosine rule formula will make the process of learning the topic at hand easier. So, with any further ado lets begin.

By the cosine trigonometric law, the side “g” will be:

g2 = h2 + k2 – 2hk cos (g)

FAQ (Frequently Asked Questions)

1. What is cosine in a right-angled triangle?

Cosine of any angle of the right-angled triangle can be calculated out by gaining knowledge on the trigonometric functions and formulas about the cosine functions. The ratio between the adjacent side of the right-angle triangle and the hypotenuse of the right-angle triangle is called the cosine function. The cosine function is included in the three main primary trigonometric functions, and it is the complement of the sine function i.e. cosine. This is a clear indication that the cosine function values will inevitably vary for different measurements of the angles formed by the sides of the triangle. It will be required frequently to find the values of cosine functions.

2. What is the domain of the cosine function?

In the cartesian point of view, as the graph of the cosine function is concerned, as we traverse towards the vertex of the point of the angle which needs to be calculated via the cosine graph. The graph shape often gets repeated often, and after every rotation of every angle, the 'periodic' functions are to be noticed. The period of each and every function is equivalent to 360 degrees or 2π radians. By this explanation, we can also equate the set and subset of values of the cosine functions. Now by using that knowledge, we can further map the functions and find the relations about the cosine functions using the Venn diagrams.