Cos function (or cosine 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:
 Cosine Definition
 Cosine Formula
 Cosine Table
 Cosine Properties With Respect to the Quadrants
 Cos Graph
 Inverse Cosine (arccos)
 Cosine Identities
 Cos Calculus
 Law of Cosines in Trigonometry
 Additional Cos Values
 Cosine Worksheet
 Trigonometry Related Articles for Class 10
 Trigonometry Related Articles for Class 11 and 12
 Other Trigonometry Related Topics
Other Trigonometric Functions
Tan Function 

Cosec (Csc) Function 
Sec Function 
Cot Function 
Cosine Definition
In a righttriangle, 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 \(\alpha\) as the angle between hypotenuse and base.
Now, for this triangle,
\(cos\, \alpha\, =\, \frac{Adjacent\,Side}{Hypotenuse}\)Cosine Formula
From the definition of cos, it is now known that it is the adjacent side devided by the hypotenuse. Now, from the above diagram,
\(cos\, \alpha\, =\, \frac{AC}{AB}\)Or,
\(cos\, \alpha\, =\, \frac{b}{h}\)Cosine Table
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 
Cosine Properties With Respect to the Quadrants
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 degrees 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 Degrees 
1st Quadrant 
+ (Positive) 
0 < cos(x) < 1 
90 to 180 Degrees 
2nd Quadrant 
– (Negative) 
1 < cos(x) < 0 
180 to 270 Degrees 
3rd Quadrant 
– (Negative) 
1 < cos(x) < 0 
270 to 360 Degrees 
4th Quadrant 
+ (Positive) 
0 < cos(x) <10 
Cos Graph
The cosine graph or the cos graph is an updown graph just like the sine graph. The only difference between sine graph and cos graph is that sine graph starts from 0 while the cos graph starts from 90 (or \(\frac{\pi}{2}\)). The cos graph given below starts from 1 and falls till 1 and then starts rising again.
Arccos (Inverse Cosine)
The cos inverse function can be used to measure the angle of any right angled triangle if the ratio of the the adjacent side and hypotenuse is given. The inverse of sine is denoted as arccos or \(cos^{1}\).
For an right triangle with sides 1, 2, and \(\sqrt{3}\), the cos function can be used to measure the angle.
In this, the cos of angle A will be, cos(a)= adjacent/hypotenuse.
So, \(cos(a)\,=\,\frac{\sqrt{3}}{2}\)
Now, the angle “a” will be \(cos^{1}(\frac{\sqrt{3}}{2})\)
Or, \(a\,=\,\frac{\pi}{6}\,=\,30\)°
Important Cos Identities
 \(cos^{2}(x)\,+\,sin^{2}(x)\,=\,1\)
 \(cos\,\theta\,=\,\frac{1}{sec\,\theta}\)
 \(cos(\theta)\,=\,cos(\theta)\)
 \(arccos(cos(x))\,=\,x\,+\,2k\pi\; where \,k=integer\)
 \(Cos(2x)=cos^{2}(x)\,\,sin^{2}(x)\)
 \(cos(\theta)\,=\,sin(\frac{\pi}{2}\,\,\theta)\)
Below, all the other trigonometric functions in terms of cos function are also give.
Other Trigonometric Functions in Terms of Sine
Trigonometric Functions 
Represented as Sine 
\(sin\,\theta\)  \(\pm \sqrt{1cos^{2}\theta}\) 
\(tan(\theta)\)  \(\pm \frac{\sqrt{1cos^{2}\,\theta}}{cos\,\theta}\) 
\(cot(\theta)\)  \(\pm\frac{cos\,\theta}{\sqrt{1\,\,cos^{2}\,\theta}}\) 
\(sec(\theta)\)  \(\pm\frac{1}{cos\,\theta}\) 
\(cosec(\theta)\)  \(\pm\frac{1}{\sqrt{1\,\,cos^{2}\,\theta}}\) 
Cos Calculus
For cosine function \(f(x)\,=\,cos(x))\), the derivative and the integral will be given as:
 Derivative of cos(x), \(f'(x)\,=\,sin(x))\)
 Integral of cos(x), \(\int f(x)\,dx\,=\,sin(x)\,+\,C)\) (where C is the constant of integration)
Law of Cosines in Trigonometry
The law of cosine or cosine rule in trigonometry is an relation between the side and the angles of a triangle. Suppose a triangle with sides a, b, c and with angles A, B, 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.
Also Check:
 Law of Sines
 Tan Law
Additional Cos Values
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 
Cos Questions (Worksheets)
 \(sin(cos^{1}\frac{3}{5})\)
 In a triangle PQR, PR is 14 cm, QR is 10 cm, and angle RQP is 63 degrees. Calculate angle RPQ and the length of PQ.
 In triangle ABC, AB 6 cm, AC is 13 cm, and angle CAB is 91 degrees. Calculate the length of BC.
 Derive the value of cos 60 geometrically.
 A ramp is pulled out of the back of a truck. There is a 38 degrees angle between the ramp and the pavement. The distance from the end of the ramp to to the back of the truck is 10 feet. Calculate the length of the ramp?
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