The cosine function (or cos function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine(co+sine). There are various topics that are included in the entire cos concept. Here, the main topics that are focussed include:
| Sine Function | Tan Function |
| Cosec (Csc) Function | Sec Function |
| Cot Function |
In a right-triangle, cos is defined as the ratio of the length of the adjacent side to that of the longest side i.e. the hypotenuse. Suppose a triangle ABC is taken with AB as the hypotenuse and α as the angle between the hypotenuse and base.
Now, for this triangle,
cos α = Adjacent Side/Hypotenuse
From the definition of cos, it is now known that it is the adjacent side divided by the hypotenuse. Now, from the above diagram,
c o s α = A C/ A B
| Cosine Degrees | Values |
| cos 0° | 1 |
| cos 30° | √3/2 |
| cos 45° | 1/√2 |
| cos 60° | 1/2 |
| cos 90° | 0 |
| cos 120° | -1/2 |
| cos 150° | -√3/2 |
| cos 180° | -1 |
| cos 270° | 0 |
| cos 360° | 1 |
It is interesting to note that the value of cos changes according to the quadrants. In the above table, it can be seen that cos 120°, 150° and 180° have negative values while cos 0°, 30°, etc. have positive values. For cos, the value will be positive in the first and the fourth quadrant.
| Degree Range | Quadrant | Cos Function Sign | Cos Value Range |
| 0°to 90° | 1st Quadrant | + (Positive) | 0 < cos(x) < 1 |
| 90° to 180° | 2nd Quadrant | – (Negative) | -1 < cos(x) < 0 |
| 180° to 270° | 3rd Quadrant | – (Negative) | -1 < cos(x) < 0 |
| 270° to 360° | 4th Quadrant | + (Positive) | 0 < cos(x) |
The cosine graph or the cos graph is an up-down graph just like the sine graph. The only difference between the sine graph and the cos graph is that the sine graph starts from 0 while the cos graph starts from 90 (or π/2). The cos graph given below starts from 1 and falls till -1 and then starts rising again.
The cos inverse function can be used to measure the angle of any right-angled triangle if the ratio of the adjacent side and hypotenuse is given. The inverse of sine is denoted as arccos or cos -1 x.
For a right triangle with sides 1, 2, and √3, the cos function can be used to measure the angle.
In this, the cos of angle A will be, cos(a)= adjacent/hypotenuse.
Now, the angle “a” will be cos −1 (√3/2)
Below, all the other trigonometric functions in terms of cos function are also given.
| Trigonometric Functions | Represented as Sine |
| sin θ | ±√(1-cos 2 θ) |
| tan θ | ±√(1-cos 2 θ)/cos θ |
| cot θ | ±cos θ/√(1-cos 2 θ) |
| sec θ | ±1/cos θ |
| cosec θ | ±1/√(1-cos 2 θ) |
For cosine function f ( x ) = c o s ( x ) , the derivative and the integral will be given as:
The law of cosine or cosine rule in trigonometry is a relation between the side and the angles of a triangle. Suppose a triangle with sides a, b, and c and with angles A, B, and C are taken, the cosine rule will be as follows.
According to cos law, the side “c” will be:
c 2 = a 2 + b 2 − 2ab cos (C)
It is important to be thorough with the law of cosines as questions related to it are common in the examinations.
| Cos 1 Degree is 0.99 | Cos 2 Degree is 0.99 |
| Cos 5 Degree is 0.996 | Cos 8 Degree is 0.990 |
| Cos 10 Degree is 0.984 | Cos 15 Degree is 0.965 |
| Cos 20 Degree is 0.939 | Cos 30 Degree is 0.866 |
| Cos 40 Degree is 0.766 | Cos 50 Degree is 0.642 |
| Cos 70 Degree is 0.342 | Cos 80 Degree is 0.173 |
| Cos 100 Degree is -0.173 | Cos 105 Degree is -0.258 |
| Cos 210 Degree is -0.866 | Cos 240 Degree is -0.5 |
| Cos 270 Degree is 0 | Cos 330 Degree is 0.866 |
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Quiz on Cosine Function
